Graph polynomials: some questions on the edge

Abstract

We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction relations (simple linear recursions based on local operations), perhaps in a wider class of combinatorial objects? How many levels of reduction relations does a graph polynomial need in order to express it in terms of trivial base cases? For a graph polynomial, how are properties such as equivalence and factorisation reflected in the structure of a graph? We illustrate our discussion with a variety of graph polynomials and other invariants. This leads us to reflect on the historical origins of graph polynomials. We also introduce some new polynomials based on partial colourings of graphs and establish some of their basic properties.

Full text

Graph polynomials: some questions on the edge
Graham Farr
Department of Data Science and AI
Faculty of I.T.
Monash University
Australia
Graham.Farr@monash.edu
Kerri Morgan
School of Science (Mathematical Sciences)
RMIT University
Australia
Kerri.Morgan@rmit.edu.au
22 June 2024
Abstract
We raise some questions about graph polynomials, highlighting concepts
and phenomena that may merit consideration in the development of a general
theory. Our questions are mainly of three types: When do graph polynomials
have reduction relations (simple linear recursions based on local operations),
perhaps in a wider class of combinatorial objects? How many levels of reduction
relations does a graph polynomial need in order to express it in terms of trivial
base cases? For a graph polynomial, how are properties such as equivalence and
factorisation reflected in the structure of a graph? We illustrate our discussion
with a variety of graph polynomials and other invariants. This leads us to
reflect on the historical origins of graph polynomials. We also introduce some
new polynomials based on partial colourings of graphs and establish some of
their basic properties.
1
arXiv:2406.15746v1 [math.CO] 22 Jun 2024

1 Introduction
J´anos Makowsky started his research career in mathematical logic over half a cen-
tury ago. For the last two decades, he has brought many concepts and results from
that field — along with the perspective it offers — to bear on the study of graph
polynomials. This has led to significant new theorems, links between topics, fresh
viewpoints, and deeper understanding.
His contributions to graph polynomials, with a wide set of collaborators, include:
complexity classifications for specific computational problems [9, 65, 68, 74] (see also
[61]); refining the models of computation used to study the computational complexity
of problems concerning graph polynomials [75, 62]; introducing formal logical defi-
nitions of graph polynomials, especially using second-order logic (SOL) and variants
thereof, and using concepts and tools from logic to develop their theory at a very
general level [18, 67, 69, 64, 48]; developing a general framework for studying the dis-
tinguishing power of graph polynomials [77, 76, 63]; and showing that the locations
of zeros of a graph polynomial is “not a semantic property”, in that it derives more
from the algebraic form of the polynomial than from the way it partitions the set of
all graphs into equivalence classes [77, 78]. Some of his own reflections on these topics
may be found in [78, 72, 73].
A pervasive characteristic of his work has been to put specific graph polynomials
in a wider mathematical context — to see the forest as well as the trees, to quote
his own quotation of Einstein [71]. One manifestation of this has been his advocacy
for the development of a general theory of graph polynomials and his own work
in that direction [69, 70, 78]. He was a co-organiser of the Dagstuhl Seminar on
“Graph Polynomials: Towards a Comparative Theory” in 2016 [23]. At the Dagstuhl
Seminar on “Comparative Theory for Graph Polynomials” in 2019 [24], he helped
lead an informal group working on the distinguishing power of graph polynomials.
In this paper we propose some polynomials, topics and questions that may merit
further consideration in the development of a general theory of graph polynomials.
To do this, we journey to the edge of the territory covered by the current theory,
meeting some polynomials that seem to lie near, or beyond, that frontier, as well as
some that are very familiar and are well covered by the theory.
We begin with some general definitions and notation in §2 and set the scene in
§3 by defining all the polynomials we will discuss. This includes the introduction
of some new graph polynomials related to partial colourings. In §4 we reflect on
the origins of graph polynomials. Some of the polynomials we introduce are then
used to illustrate the questions raised in the next four sections. In §5 we consider
the widespread phenomenon of graph polynomials having reduction relations (i.e.,
simple recursive relations based on local modifications), pointing out that even graph
polynomials that do not seem to have such a relation will often be found to have one
within a wider class of objects. In §6 we discuss a notion of “levels” in these reduction
relations, where a graph polynomial can be reduced to a large sum of polynomials of
reduced objects of some kind, which in turn may each be reduced to another large
2

sum using another relation, and so on. In §7 we discuss the relationship between the
algebraic properties of a graph polynomial and the structure of the graph. In §8 we
pose questions about certificates, a tool for studying equivalence and factorisation of
graph polynomials.
Some of the material in this paper was presented by the first author in a talk of the
same title at the Dagstuhl Seminar on “Graph Polynomials: Towards a Comparative
Theory” in June 2016 [23].
2 Definitions and notation
Throughout, G= (V, E ) is a graph with nvertices and medges. The number of
components of Gis denoted by k(G). If X⊆Ethen V(X) denotes the set of
vertices of Gthat are incident with at least one edge in X(overloading the V( )
notation slightly). We write ν(X) for the number of vertices of Gthat meet an
edge in X(succinctly: ν(X) = |V(X)|). The number of components of (V, X ) is
denoted by k(X), while the number of components of (V(X), X) is denoted by κ(X).
The former count includes isolated vertices, while the latter count excludes them:
k(X) = κ(X) + n−ν(X). We write ρ(X) for the rank of X, given by ρ(X) =
ν(X)−κ(X) = |V| − k(X), and ρ(G) := ρ(E(G)). For any function r: 2E→R,
its dual r∗is defined by r∗(X) = |X|+r(E\X)−r(E) + r(∅). When ρis the rank
function of G(and therefore the rank function of its cycle matroid), ρ∗is the rank
function of the dual of the cycle matroid of G.
If U⊆Vthen G[U] is the subgraph of Ginduced by U.
The disjoint union of two graphs Gand His denoted by G⊔H.
If e∈Ethen G\e= (V, E \ {e}) is the graph obtained from Gby deleting
edge eand G/e is the graph obtained by contracting edge e, i.e., deleting eand then
identifying its former endpoints. If u, v ∈Vwith uv ∈ Ethen G+uv = (V, E ∪ {uv})
is the graph obtained by adding an edge between uand vin Gand G/uv is obtained
from Gby identifying vertices uand v.
Acoloop of Gis an edge esuch that k(G\e)> k(G). This is often called a bridge
and sometimes an isthmus.
The maximum degree of Gis denoted by ∆(G).
Anull graph is a graph with no edges.
Agraph invariant is a function defined on all graphs that depends only on the
isomorphism class of the graph.
We write [k] = {1,2, . . . , k}.
The falling factorial x(x−1) · · · (x−k+ 1) is denoted by (x)k.
Let Λ be a set whose members we will call colours, and let λ∈N. A Λ-assignment
of Gis a function f:V→Λ. A λ-assignment is a [λ]-assignment. A partial Λ-
assignment is a function f:W→Λ where W⊆V. The vertices of Wand V\W
are coloured and uncoloured, respectively, by f. A partial λ-assignment is a partial
[λ]-assignment. For every k∈Λ, its colour class C(k) = Cf(k) under a partial Λ-
assignment fis given by C(k) = f−1(k) = {v∈V:f(v) = k}. Every colour class
3

C(k) induces a subgraph G[C(k)] of G. A chromon, or monochromatic component, of
(G, f ) is a component of G[C(k)] for some k∈[λ].
Every partial λ-assignment fis determined by its λ-tuple (Cf(i))λ
i=1 of colour
classes: given a λ-tuple (Ci)λ
i=1 of mutually disjoint subsets of V, we can define a
partial λ-assignment f:Sλ
i=1 Ci→[λ] by f(v) = ifor each v∈Ciand i∈[λ]; this
then satisfies Cf(i) = Cifor all i∈[λ].
A colour class is proper if it is a stable set in G, i.e., no two of its vertices are
adjacent in G. A partial λ-assignment is a partial λ-colouring if every colour class is
proper; this is regardless of whether or not the partial λ-assignment can be extended
to a λ-colouring of G.
An extension of a partial λ-assignment fis a partial λ-assignment gsuch that
f⊆g, which is equivalent to requiring that dom f⊆dom gand f(v) = g(v) for all
v∈dom f.
3 Some graph polynomials
Papers by J´anos often include observations about collections of specific graph poly-
nomials as motivation for studying more general phenomena. In a similar spirit, we
now discuss an eclectic collection of graph invariants — some old, some new; mostly
polynomials, some not — to help motivate some of the questions we ask. The reader
can skip those sections treating graph polynomials with which they are familiar.
More comprehensive collections of graph polynomials may be found in [26, 103].
3.1 Tutte-Whitney polynomials
The preeminent graph polynomial is arguably the Tutte polynomial, due to W.T.
Tutte [106, 107] and closely related to the Whitney rank generating function [116].
The Whitney rank generating function R(G;x, y) of a graph Gis the bivariate
polynomial defined by
R(G;x, y) = X
X⊆E
xρ(E)−ρ(X)yρ∗(E)−ρ∗(E\X)=X
X⊆E
xρ(E)−ρ(X)y|X|−ρ(X)
The Tutte polynomial T(G;x, y) may be defined by
T(G;x, y) = 






1,if Ghas no edges;
x T (G\e;x, y),if eis a coloop in G;
y T (G/e;x, y),if eis a loop in G;
T(G\e;x, y) + T(G/e;x, y),otherwise,
(1)
for any edge e∈E(G). The polynomials of Whitney and Tutte are related by a
simple coordinate translation: T(G;x, y) = R(G;x−1, y −1) [106, 107].
The recurrence (1) is the most fundamental property of the Tutte polynomial, and
one of Tutte’s major conceptual contributions in [106] was to base the theory on this
4

Figure 1: The two Gray graphs, from [109].
relation, a fundamental conceptual advance upon the pioneering work of Whitney
[116]. For a history of these polynomials, see [40].
A graph Gis Tutte equivalent to a graph Hif T(G;x, y) = T(H;x, y) [109, 116,
117]. Graphs with isomorphic cycle matroids are Tutte equivalent, since the Tutte
polynomial of a graph depends only on its cycle matroid. But, as an equivalence
relation on graphs, Tutte equivalence is coarser than cycle matroid isomorphism.
Tutte [109] gave two graphs, due to M.C. Gray, which are not isomorphic, and do not
even have isomorphic cycle matroids, but which have the same Tutte polynomial: see
Figure 1. Many other such pairs are known.
The most famous specialisation of the Tutte polynomial is the chromatic polyno-
mial P(G;q) = (−1)ρ(G)qk(G)T(G;−q+ 1,0) introduced by Birkhoff in [8]. For q∈N,
it gives the number of q-colourings of G. Two graphs are chromatically equivalent if
they have the same chromatic polynomial.
Tutte-Whitney polynomials have been generalised from graphs to many other
combinatorial objects [26, 33]. For structures on which deletion and contraction
operations exist, Krajewski, Moffatt and Tanasa [66] show how to use Hopf algebras
to define a polynomial that may reasonably be called the Tutte polynomial for those
structures and which satisfies a deletion-contraction relation.
3.2 Some partition functions
We introduce the partition functions of three interaction models on graphs: the Ising,
Potts and Ashkin-Teller models. Our main focus later will be on the Ashkin-Teller
model.
The set of edges of a graph Gwhose endpoints receive the same colour under
aq-assignment fis denoted by E+(f); these edges are sometimes called bad since
they are not properly coloured in the sense of graph colouring. The set of edges
whose endpoints are differently coloured under fis denoted by E−(f); these edges
are sometimes called good. Note that E+(f)∪E−(f) = E. So the positive and
negative signs in superscripts here represent “bad” and “good”, respectively, which
is the opposite of their connotations in ordinary English usage.
With this notation, we may write the chromatic polynomial as
P(G;q) = X
f:V→[q]
0|E+(f)|1|E−(f)|
5

where 0kis taken to be 1 if k= 0 and 0 otherwise, so that only proper colourings
contribute to the sum, with all proper colourings counted once.
Suppose now that we do not penalise improper colourings so drastically, but just
weight colourings according to an exponential function of the number of good edges
they have. This gives the Potts model, introduced in [96] and generalising the q= 4
case introduced by Ashkin and Teller in [2]. (See [40, §34.12]. The model is called
the Ashkin-Teller-Potts model in [114, §4.4].) The Potts model partition function is
given by1
ZPotts(G;K, q) = X
f:V→[q]
e−K·|E−(f)|.
This is a polynomial in e−Kand is known to be a partial evaluation of the Tutte
polynomial [27, 46]. The relationship between them can be expressed more simply in
terms of the Whitney rank generating function:
ZPotts(G;K, q) = qk(G)(eK−1)ρ(G)e−K|E|R(G;q
eK−1, eK−1).
The q= 2 case of the Potts model partition function is mathematically almost
the same as the Ising model partition function [58]. In the Ising model, colours take
values in {±1}, and we sum over all {±1}-assignments σ:V→ {±1}. For a given
σ:V→ {±1}, the edge uv belongs to the set Eσ(u)σ(v)(f). The Ising model partition
function is
ZIsing(G;K) = X
σ:V→{±1}
eKPuv∈Eσ(u)σ(v)
=X
σ:V→{±1}
eK·|E+(σ)|−K·|E−(σ)|
=eK|E|X
σ:V→{±1}
e−2K·|E−(σ)|
=eK|E|ZPotts(G; 2K, 2).
The Ashkin-Teller model [2] extends the Ising model and the q= 4 case of the
Potts model. Each vertex v∈Vhas a pair of colours σ(v) and τ(v), each taking
one of the two values ±1, so the available colour pairs (σ(v), τ (v)) for each vertex
are the four pairs (±1,±1). We think of the {±1}-assignments σ:V→ {±1}and
τ:V→ {±1}as two Ising spin functions on G. An example is given in Figure 2,
where the spins at upper left and lower right of each vertex are those assigned by
σand τ, respectively. These spins are also shown as colours on the left and right
semicircles in each vertex.
These yield a third such function, their product στ :V→ {±1}, given for
each v∈Vby (στ)(v) = σ(v)τ(v). So we have three Ising spin functions, coupled
1Sometimes, it is defined instead as Pf:V→[q]eK·|E+(f)|, which may be written
eK|E|Pf:V→[q]e−K·|E−(f)|. So the only change is an extra prefactor eK|E|. See, e.g., [5, §2].
6

+1
−1
−1
−1
+1
+1
−1
+1
Figure 2: An Ashkin-Teller model configuration (σ, τ) for a graph.
+1
−1
−1
−1
+1
+1
−1
+1
Left colours, σ
+1
−1
−1
−1
+1
+1
−1
+1
Right colours, τ
+1
−1
−1
−1
+1
+1
−1
+1
Product colours, στ
Figure 3: Good (thick, black) and bad (thin, red) edges for the three Ising-type
configurations in the Ashkin-Teller model configuration in Figure 2.
in the sense that they are not all independent: we can regard any two of them as
independent, but together they determine the third. Each of them gives its own
division of E(G) into good and bad edges, so we have three such divisions altogether:
spin edge type
good bad
σ E−(σ)E+(σ)
τ E−(τ)E+(τ)
στ E−(στ )E+(στ )
We illustrate these divisions, for the graph and configuration of Figure 2, in Figure 3.
The partition function of the symmetric Ashkin-Teller model is given by
ZSymAT(G;K, K′) = e(2K+K′)|E|X
σ,τ:V→{±1}
e−2K·|E−(σ)|−2K·|E−(τ)|−2K′·|E−(στ )|.
The symmetry is because the coefficients of |E−(σ)|and |E−(τ)|in the exponent are
the same, namely K. In the general four-state Ashkin-Teller model, these can be
7

different, and the partition function depends on three variables instead of the two
(K,K′) we have here.
If K′= 0 then ZSymAT(G;K, K′) is just the square of an Ising model partition
function:
ZSymAT(G;K, 0) = ZIsing(G;K)2.
If K=K′then ZSymAT(G;K, K′) is just a Potts model partition function up to a
simple factor:
ZSymAT(G;K, K) = e−3K|E|ZPotts(G; 4K, 4).
In these two cases, the symmetric Ashkin-Teller model partition function can
be obtained from the Tutte polynomial, since the Ising and Potts model partition
functions can be so obtained. But this is not possible in general. Direct evaluation
shows that the two Tutte equivalent Gray graphs (Figure 1) have different symmetric
Ashkin-Teller model partition functions. From this we see that the symmetric Ashkin-
Teller model partition function is not a specialisation of the Tutte polynomial.
3.3 Interpolating between contraction and deletion
We can represent any subset X⊆Eof the edge set of a graph by its characteristic
vector x= (xe)e∈E∈GF(2)Edefined by
xe=1,if e∈E,
0,if e∈ E.
Acocircuit of Gis a minimal set of edges whose removal does not disconnect any
component of G. These are also the circuits of the dual of the cycle matroid of G.
The cocircuit space of Gis the linear space over GF(2) generated by the characteristic
vectors of cocircuits of G. This is also the row space of the binary incidence matrix
of G, which has rows indexed by V, columns indexed by E, and each entry is 1 or 0
according as its vertex is, or is not, incident with its edge.
In [29], the operations of contraction and deletion are applied directly to the
indicator functions of cocircuit spaces. Let f: 2E→ {0,1}be the indicator function
of the cocircuit space of G. Let f//e : 2E→ {0,1}and f\\e: 2E→ {0,1}be the
indicator functions of G/e and G\e, respectively. Then it is shown in [29] that
(f//e)(X) = f(X)
f(∅),(f\\e)(X) = f(X) + f(X∪ {e})
f(∅) + f({e}).
This is used in two related generalisations. Firstly, in [29], contraction and deletion are
extended to arbitrary f: 2E→Rsatisfying f(∅) = 1; in later work it was convenient
to include all f: 2E→C. Such functions fare called binary functions, and if fis the
indicator function of the cocircuit space of a graph then it is graphic. Secondly, in [32],
contraction and deletion are considered to be just two specific reductions in a whole
family of λ-reductions parameterised by λ∈R, and later by λ∈Cin [34], using the
8

notation and expression in the middle column, where e∈Eand X⊆E\ {e}:
Contraction λ-reduction Deletion
(λ= 0) (λ= 1)
(f//e)(X) (f∥λe)(X) (f\\e)(X)
f(X)
f(∅)
f(X) + λf(X∪ {e})
f(∅) + λf({e})
f(X) + f(X∪ {e})
f(∅) + f({e})
Ordinary contraction and deletion are given by λ= 0 and λ= 1, respectively. When
λ∈[0,1], one can think of the λ-reduction as interpolating between contraction and
deletion. These λ-reductions come in dual pairs, with the dual of λ-reduction being
λ∗-reduction where
λ∗=1−λ
1 + λ.
When λ∈ {0,1}, a λ-reduction of a graphic binary function is not, in general, graphic.
These dual λ-reduction operations are accompanied by parameterised versions of
the rank function and Whitney rank generating function. The λ-rank function Q(λ)f
is defined by
(Q(λ)f)(X) = log2 (1 + λ∗)|V|PW⊆Eλ|W|f(W)
PW⊆Eλ|W∩(E\X)|(λ∗)|W∩X|f(W)!,
where X⊆E. The λ-Tutte-Whitney function is defined by
R(λ)
1(G;x, y) = R(λ)
1(f;x, y) = y−Q(λ)f(E)X
X⊆E
(xy)Q(λ)f(E)−Q(λ)f(X)y|X|.
These functions are shown in [32] to satisfy a generalisation of the contraction-deletion
relation of the same form as for Tutte-Whitney polynomials, with λ-reductions and
their dual λ∗-reductions taking the place of contraction and deletion. The loopiness
and coloopiness of the element e∈Sunder the function fare defined by the functions
loop(λ)(f, e) = Q(λ∗)f(E)−Q(λ∗)f(E\e),
coloop(λ)(f, e) = Q(λ)f(E)−Q(λ)f(E\e) = loop(λ∗)(f, e).
Theorem 1 ([32]) For any binary function f: 2E→Cand any e∈E,
R(λ)(f;x, y) = xcoloop(λ)(f ,e)R(λ)(f∥λe;x, y) + yloop(λ)(f,e)R(λ)(f∥λ∗e;x, y).
3.4 Go polynomials
J´anos is a keen Go player. He and the first author have played several times; as the
stronger player, J´anos plays with the white stones and gives a handicap. So it is
9

fitting to mention some surprising points of contact between graph polynomials and
the theory of Go.
Here we use the name by which the game is known in Japan and in the West. It
is called W´eiq´ı in China and Baduk in Korea.
Go is normally played on the square grid graph of 19×19 vertices. But it has long
been recognised that Go is an entirely graph-theoretic game and can be played on
any graph. It has been played, for example, on a three-dimensional diamond lattice
graph [98] and on a map of Milton Keynes [57].
Thorpe and Walden [101, 102] seem to have been among the first to formalise
the rules and concepts of Go in order to support mathematical and computational
investigation. Their work uses some graph-theoretic concepts but is still embedded
in rectangular grid graphs. Benson [6] uses graph-theoretic language and concepts to
characterise unconditional life of groups of stones. Tromp and Taylor [105, 99] defined
“logical rules” for Go, based on rules from the New Zealand Go Society. These rules
are concise, precise, elegant, and purely graph-theoretic. Although they state initially
that Go is played on a 19×19 square grid, these rules may be used with only cosmetic
modifications to play on any graph.
Let Gbe a graph and fbe a partial λ-assignment of G. A chromon of (G, f ) is
free if it has a vertex that is adjacent to an uncoloured vertex. A partial λ-assignment
fis a legal λ-position in G, or just a legal position if λis clear from the context, if
every chromon is free.
Normally, Go is played with just two colours, Black and White, so λ= 2, although
multiplayer versions with λ > 2 have been proposed and equipment (in the form of
coloured Go stones) is available for them.
In Go terminology, chromons correspond to what a Go player might call chains.
A chromon is free if it is a chain with at least one liberty, and a legal position is one
in which every chain has a liberty. When playing the game according to the rules,
every position will be a legal position except during the brief intervals after a capture
is made and before all the captured stones are removed from the board. We do not
discuss the rules of Go further; see, e.g., [59] for more information.
Two Go polynomials based on these concepts were introduced in [31], studied
further in [42, 36, 39], and mentioned by J´anos in his first inventory of the Zoo
[69, 70]. One of them simply counts legal positions:
Go#(G;λ) = number of legal λ-positions for G.
For example, it can be shown that Go#(C4;λ) = 1 + 14λ2. The other Go polynomial
from [31] is based on a simple probability model. Let p≤1
2and construct a random
partial 2-assignment fas follows. Each v∈Vis assigned colour 1 or 2, with prob-
ability peach, or is left uncoloured with probability r:= 1 −2p, with decisions for
different vertices being independent. Under this model, define
Go(G;p) = Pr(fis a legal 2-position for G).
We could also define a bivariate version of the second polynomial by extending the
probability model to λcolours, where fis now a partial λ-assignment, the probability
10

pnow satisfies p≤λ−1, and fis assigned colour k∈[λ] with probability pfor each
colour and is left uncoloured with probability r:= 1 −λp:
Go(G;p, λ) = Pr(fis a legal λ-position for G).
All three functions can be shown to be polynomials, so they can take as arguments
any λ, p ∈Calthough we only know of combinatorial interpretations for the values
used in the definitions above. This suggests the problem of finding combinatorial
interpretations at other values of pand λ. We suggest λ=−1 as one that might
be worth exploring, since the chromatic polynomial has an interesting combinatorial
interpretation at λ=−1, namely the number of acyclic orientations [100], and Go
polynomials can be expressed naturally as sums of chromatic polynomials [31].
These Go polynomials are all exponential-time computable, and they seem un-
likely to be polynomial-time computable because it was shown in [31, §4] that, for
any fixed integer λ≥2, computing the value of Go#(G;λ) for an input graph Gis
#P-hard, using methods from transcendental number theory.
3.5 Polynomials for partial colourings
Given a graph G, a nonnegative integer λrepresenting some number of available
colours, and a probability p≤λ−1, put r:= 1 −λp and define the following random
colouring model, noting that the partial λ-assignment it produces is not necessarily
a (proper) colouring.
Each vertex v∈V(G) remains uncoloured with probability rand otherwise is
given a colour chosen uniformly at random from the λavailable colours. So, for
any specific colour, the probability that vgets that colour is p. The choices made
at different vertices are independent. This process generates a random partial λ-
assignment whose domain is a subset of V(G).
A partial λ-assignment fof Gis a partial λ-colouring if it is a colouring of
G[dom f], the subgraph of Ginduced by the coloured vertices. In a partial λ-
colouring, vertices that are adjacent in Gcannot both get the same colour; either
they receive different colours or at least one of them is uncoloured.
We say fis λ-extendable if there is a λ-colouring of Gthat extends f.
A vertex v∈ dom fis immediately λ-forced by fif its neighbours in dom fhave
exactly λ−1 distinct colours among them, and in this case we write f;vfor the
partial λ-assignment on (dom f)∪ {v}which agrees with fon dom fand gives vthe
sole colour from [λ] that does not appear among its neighbours. The motivation is
that, if vertex vis to receive a colour from [λ] without creating any bad edges, then it
must be given that one colour that is unused by any of its neighbours. If the number
of different colours that appear among the neighbours of vis λ, then there is no hope
for v: any colouring of it creates one or more bad edges. If the neighbours of vonly
have ≤λ−2 colours, then the colour of vis not determined by the colours of its
neighbours, as there are at least two options for it.
If fimmediately λ-forces v, then there is only one possible colour for vin all λ-
colourings of Gthat extend f. But the converse does not hold in general: the colour
11

of vmay be uniquely determined without being forced in this specific local sense.
The vertex vis (eventually) λ-forced by fif there is a sequence of vertices v1, . . . ,vk=
v, all outside dom f, and a sequence of partial λ-assignments f=f0, f1, . . . , fksuch
that, for all i∈[λ],
•dom fi= (dom f)∪ {v1, . . . , vi},
•viis immediately forced by fi−1,
•fi=fi−1;vi.
We may drop λfrom “λ-forced” when it is clear from the context.
Note that if fis improper then it is not λ-extendable, and that if fis not λ-
extendable then (regardless of whether or not it is proper) it cannot force a λ-colouring
of G.
For every graph Gwe define three bivariate functions based on partial colourings.
The partial chromatic polynomial PC(G;p, λ) is defined by
PC(G;p, λ) = Pr(fis a λ-colouring of G[dom f]),
where p∈[0, λ−1] is the probability that a specific vertex gets a particular colour,
and λ∈N. Since PC(G;p, λ) is a polynomial (as we will shortly see), its domain
is C2. Observe that PC(G;λ−1, λ) = λ−nP(G;λ) and PC(G; 0, λ) = 1. The partial
chromatic polynomial is a simple algebraic transformation of the generalised chromatic
polynomial P(G;x, y) introduced by [22]. For x∈Nand y∈[x], the value of
P(G;x, y) is the number of [x]-assignments for which the first ycolour classes are
proper. If we divide by the total number xnof all [x]-assignments, then this falls
within the random partial colouring framework by putting p=x−1,λ=yand
r= 1 −x−1y. So we have
P(G;x, y) = PC(G;x−1, y)
xn.
The change of variables is easily inverted to obtain
PC(G;p, λ) = P(G;p−1, λ)
pn.
By considering all possibilities for the domain of f, we obtain
PC(G;p, λ) = X
C⊆V
P(G[C]; λ)p|C|(1 −λp)n−|C|,(2)
which is just a rearrangement of [22, Theorem 1] and shows that PC(G;p, λ) is indeed
a polynomial. When λis fixed, we denote the resulting polynomial in pby PCλ(G;p).
The partial chromatic polynomial is also a specialisation of some trivariate graph
polynomials that count edge-subsets according to (|X|, ν(X), κ(X)) or according to
12

simple invertible transformations of them, e.g., (|X|, ν(X), ρ(X)). Historically, the
first of these trivariate polynomials seems to have been the Borzacchini-Pulito poly-
nomial [11], given by
BP(G;x, y, z) = X
X⊆E
x|X|yk(X)zνX =X
X⊆E
x|X|yκ(X)+n−ν(X)zνX =X
X⊆E
x|X|yn−ρ(X)zνX .
They gave a ternary reduction relation for it in [11, Theorem 2]. Averbouch, Godlin
and Makowsky [3, 4] introduced the edge elimination polynomial ξ(G;x, y, z ) as the
most general trivariate graph polynomial that satisfies a ternary reduction relation
using edge deletion, contraction and extraction (deletion of an edge, its endpoints and
their incident edges) and showed how to express it as a sum over pairs of disjoint sub-
sets of edges. Trinks [104] showed that these two trivariate polynomials are equivalent
in the sense that each can be transformed to the other by simple transformations of
the variables.
The extendable colouring function is given by
EC(G;p, λ) = Pr(fis λ-extendable).
Observe that EC(G;λ−1, λ) = λ−nP(G;λ) and EC(G; 0, λ) is 1 if Gis 3-colourable
and 0 otherwise. We can express EC(G;p, λ) as a sum over all possible domains for
f,
EC(G;p, λ) = X
C⊆V
EC(G, C;λ)p|C|(1 −λp)n−|C|,
where EC(G, C;λ) is the number of partial λ-colourings fsuch that domf=Cand f
has an extension which is a λ-colouring of G. But this does not show that EC(G;p, λ)
is a polynomial. In fact, in general it is not. Consider K2.
EC(K2;p, λ) = 0,if λ= 1,
1−λp2,if λ≥2.(3)
It follows from (3) that EC(K2;p, λ) is not a polynomial, since when p= 0 it is 1 for
every positive integer λ≥2 but is 0 for λ= 1, a property that no polynomial can
have. Nonetheless, EC(G;p, λ) is a polynomial in pfor every fixed λ∈N. When λis
fixed, we denote this polynomial in pby ECλ(G;p).
For the extendable colouring polynomial ECλ(G;p), even checking the structures
being counted seems hard in general for λ≥3. Testing whether a given partial λ-
assignment of a given graph Gis extendable to a λ-colouring of Gis NP-complete
when λ≥3, by polynomial-time reduction from graph λ-colourability: when the
partial λ-assignment is the empty λ-assignment, leaving all vertices uncoloured, we
just have standard λ-colourability. So this polynomial is likely to be more difficult to
work with than many that have been studied.
The forced colouring function is given by
FC(G;p, λ) = Pr(feventually forces a λ-colouring of G).
13

Again, we have FC(G;λ−1, λ) = λ−nP(G;λ). We can express FC(G;p, λ) as a sum,
FC(G;p, λ) = X
C⊆V
FC(G, C;λ)p|C|(1 −λp)n−|C|,
where FC(G, C;λ) is the number of partial λ-colourings fsuch that dom f=Cand
feventually forces a λ-colouring of G. But, again, we do not have a polynomial, in
general. Consider a null graph.
FC(Kn;p, λ) = 1,if λ= 1,
(λp)n,if λ≥2.(4)
This is because when there is only one colour, the colour of every initially-uncoloured
isolated vertex is forced, while if there are at least two colours, an uncoloured isolated
vertex cannot be forced, so it can only get a colour from the initial random partial
colouring. It follows from (4) that FC(Kn;p, λ) is not a polynomial, since when p= 0
it is zero for every positive integer λ≥2 but is 1 for λ= 1. But FC(G;p, λ) is a
polynomial in pfor every fixed λ∈N, denoted by FCλ(G;p).
Our three bivariate functions satisfy
FC(G;p, λ)≤EC(G;p, λ)≤PC(G;p, λ)
whenever λ∈Nand 0 ≤p≤λ−1.
For these three functions, the case λ= 3 is of particular interest:
PC3(G;p) = PC(G;p, 3),
EC3(G;p) = EC(G;p, 3),
FC3(G;p) = FC(G;p, 3).
We are particularly interested in FC3(G;p) which we call the forced 3-colouring poly-
nomial of G. We list some basic examples and observations.
FC3(Kn;p) = (3p)n,
FC3(K2;p) = 6p2,
FC3(K3;p) = 6p2(3 −8p),
FC3(K1,2;p) = 6p2(1 −p),
FC3(G;1
3) = 3−nP(G; 3),
4n·FC3(G;1
4) = # partial 3-assignments that eventually force a λ-colouring of G,
FC3(G⊔H;p) = FC3(G;p)FC3(H;p).
In many graph polynomials, the structures being counted can be checked very
quickly in parallel. Typically, these checking problems belong to the class NC of de-
cision problems solvable by uniform families of logical circuits of polynomial size and
polylogarithmic depth, both of these being upper bounds in terms of the input size.
14

(See, e.g., [51] for the theory of NC and P-completeness.) For example, validity of
a particular 3-colouring, or an independent set, or a clique, is just a conjunction of
local conditions based on the edges of the graph; validity of a flow or a matching or
a dominating set is a conjunction of local conditions based on the vertex neighbour-
hoods.
One motivation for studying the forced 3-colouring polynomial is because of the
computational complexity of checking the structures being counted, i.e., checking if a
partial 3-assignment forces a 3-colouring of the graph. This can be done in polynomial
time, so the situation is not as bad as for extendable colouring polynomials, and
may seem more akin to classical graph polynomials based on enumerating structures
like colourings, independent sets, matchings and so on. But the forced 3-colouring
polynomial has the distinctive feature that its checking problem is, in a precise sense,
as hard as any problem in P. Specifically, it is logspace-complete for P [30], which is
considered to be evidence that there is no fast parallel algorithm for this test and that
it does not belong to NC. Intuitively, this may make these graph polynomials more
difficult to compute and to investigate than the many others where the structure-
checking can be done in NC. So the study of them might shed light on some less-
explored regions of the theory of graph polynomials.
The forced 3-colouring polynomial does fall within the class of SOL-definable
graph polynomials, introduced by Makowsky and colleagues [67, 70] and explained in
more detail in [48, 60]. To see this, we can express it as
FC3(G;p) = X
(C1,C2,C3)∈FC3(G)
pP3
i=1 |Ci|(1 −3p)n−P3
i=1 |Ci|,
where FC3(G) is the set of mutually disjoint triples (C1, C2, C3) whose corresponding
partial 3-assignment f— i.e., the partial 3-assignment fdefined by Cf(i) = Cifor
i∈ {1,2,3}— forces a 3-colouring of G. The condition that (C1, C2, C3)∈ FC3(G)
can be expressed in SOL using the following observation.
Proposition 2 A partial λ-assignment fforces a λ-colouring of Gif and only if it
has no extension gwhich is not total and forces no vertex outside dom g.
Proof. (=⇒) Suppose fforces a λ-colouring of g. Then every extension of fforces
the same λ-colouring of G. So there is no non-total extension of fthat does not force
any vertex.
(⇐=) If fdoes not force a λ-colouring of G, then the forcing process must stop
with at least one vertex uncoloured. When that happens, the partial λ-assignment
that has been forced so far is an extension of fthat is not total and forces no vertex
outside its domain.
This observation justifies the following logical formulation of FC3(G), which is
15

similar in design to the SOL expression for connectedness in [48, §3.3].
(C1, C2, C3)∈ FC3(G)⇐⇒
¬∃(D1, D2, D3) :
3
^
i=1
(Ci⊆Di)!∧ 3
^
i=1
3
^
j=i+1
(Di∩Dj=∅)!∧(D1∪D2∪D3=E)∧
∀v∈E\(D1∪D2∪D3) :
3
^
i=1
(∃w∈Di:vw ∈E)!∨
3
_
i=1
3
_
j=i+1
(∀w∈V:vw ∈E→(w∈ Di∪Dj))!.
This can all be expressed using the formalism of SOL, with sets of vertices represented
as unary relations, adjacency as a binary relation, and so on.
We study the forced colouring function of a graph further, along with its associated
polynomials, in [41].
4 Origins of graph polynomials
Having discussed a variety of graph polynomials in the previous section, it is a good
time to review the origins of graph polynomials in general.
Historically, graph polynomials have been created in several different ways.
(i) Sometimes, sequences of numbers that count structures of interest are used as
coefficients to define a generating polynomial for them. For example, polyno-
mials that count independent sets [55, 52], cliques [45, 119, 120], matchings
[54, 44, 49] and dominating sets [1] arose in this way.
(ii) Sometimes (but less commonly), a sequence of numbers that count structures
of interest in a graph is taken to give the values of a function of the graph and
an integer parameter, and the function turns out to be a polynomial in that
parameter. This is how the chromatic polynomial arose [8].
(iii) Sometimes, a probability model is defined on graphs, with independent, identically-
distributed random choices being made throughout the graph (typically on ver-
tices or edges) according to some probability p, and the probability that this
gives a particular type of structure is a polynomial in p. Polynomials defined this
way include the all-terminal network reliability polynomial [112] and the stabil-
ity polynomial [55, 28], as well as the polynomials based on partial colourings
that we introduced in §3.5. Such polynomials are often easily transformed into
generating polynomials, as is the case for the relationship between the stability
and independent set polynomials.
(iv) Sometimes, a graph invariant of physical interest is defined which turns out to
be a polynomial, possibly after an algebraic change of variables, as in the case of
16

the partition functions of the Ising model [58], Ashkin-Teller model [2] and Potts
model [96]. There is some overlap between this type and some previous types,
e.g., the matching polynomial is framed as a partition function in [54], and the
reliability polynomial models the survivability of networks in the presence of
local link failures.
(v) Sometimes, certain graph invariants are found to satisfy reduction relations of
some kind, and this motivates the development of a common generalisation
which turns out to be a polynomial. This is how the Tutte polynomial was
created, as Tutte relates in [110, p. 53] and [111].
(vi) Sometimes, a polynomial is created by analogy with existing polynomials and/or
by generalising them. See [33, §3.4] for a discussion of the different ways in
which Tutte-Whitney polynomials have been generalised; the informal classifi-
cation given there could apply to analogues and generalisations of any graph
polynomial. An early example of this is the Whitney rank generating function
itself, which arose in 1932 as a bivariate generalisation of the chromatic polyno-
mial [116] without noting any other combinatorial interpretations of its values
and with the deletion-contraction relation only being a “note added in proof”
and attributed to R. M. Foster. The Oxley-Whittle polynomial [93, 94] arose as
the analogue of Tutte-Whitney polynomials for 2-polymatroids. Tutte-Whitney
polynomials of graphs were also the inspiration for the various topological Tutte
polynomials for ribbon graphs and embedded graphs [17, 25].
(vii) Occasionally, a polynomial is defined by specialisation from an existing polyno-
mial, instead of by generalisation: it is not always the case that the particular
motivates the general. The flow polynomial was first discovered (but not named)
by Whitney [117] as the dual of the chromatic polynomial and a specialisation
of his rank generating function, but without giving its values any combinatorial
interpretation; see [40, §34.7]. Only later did Tutte identify the connection with
flows [107].
(viii) When counting graphs and other combinatorial objects up to symmetry, cycle
index polynomials are used: see [53] for their use in counting various types of
graphs up to isomorphism, a field which originated with Redfield [97] and P´olya
[95]. Beginning with work by Cameron and collaborators in the early 2000s,
cycle index polynomials have been used to define graph polynomials that count
colourings and other structures up to symmetries under the automorphism group
of the graph [13, 14, 15].
(ix) Sometimes, the characteristic polynomial of a square matrix associated with
a graph is studied, the main example being the characteristic polynomial of a
graph2which is obtained from the adjacency matrix.
2not to be confused with the characteristic polynomial of a matroid, which generalises the chro-
matic polynomial of a graph.
17

(x) Sometimes, a polynomial is defined for other mathematical objects, and a natu-
ral construction of graphs from those objects leads to a graph polynomial. Some
knot polynomials have been translated to graph polynomials and linked to the
Tutte polynomial: see, e.g., [56]. The weight enumerator of a linear code over
GF(2) may be regarded as a binary matroid polynomial whose coefficients count
members of a circuit space according to their size (and similarly for a cocircuit
space), thereby yielding a graph polynomial through the special case of graphic
matroids (see, e.g., [113, §15.7]).
This classification is just a set of observations of how the field has developed so far
and is certainly not any kind of prescription for how it must develop in the future.
It may not be exhaustive, and the categories may overlap. Indeed it is common for
a graph polynomial to have multiple different formulations, so that it can be defined
in two or more of the above ways, even if it was historically created in just one way.
There is no single best route to defining graph polynomials: all the routes listed above
have led to new polynomials that have generated a lot of interest and some profound
research.
Nonetheless, it is worth reflecting on the various inspirations for the diverse graph
polynomials that have been developed. Analogy and generalisation have been very
fruitful, but it is arguable that the real worth of a new graph polynomial lies not
so much in how well its theory echoes those of other known polynomials, but rather
in the information it contains about the graph (including the relationships it reveals
between different features of it) and in the accessibility of that information.
In this context, the originality of Tutte’s approach (v) is striking. It did not really
fall within any earlier approach. It was grounded in important graph invariants. His
polynomial emerged naturally as a framework that captured what those invariants
had in common. The fact that it is a polynomial was a happy outcome, and not
surprising since one of the invariants he was abstracting from was the chromatic
polynomial, but it does not seem to have been an objective in itself.
It is conceivable that other collections of enumerative (or probabilistic) graph
invariants are waiting to be brought into common frameworks, and that these frame-
works may sometimes be quite different to Tutte’s, and may not always lead to poly-
nomials.
5 Reduction relations
The recursive relation (1) for the Tutte polynomial has the following characteristics.
•The number of cases is fixed (i.e., independent of the size of the graph).
•In the base case, the expression is a fixed polynomial (in this case, just the
constant 1).
•In each other case, the expression is linear (treating the Tutte polynomial sym-
bols T(G;x, y), T(G\e;x, y), T(G/e;x, y) as indeterminates, with coefficients
18

being fixed polynomials in x, y).
•The graphs appearing in the right-hand sides are obtained from the original
graph Gby simple local operations.
•Certain types of edges must be treated as special cases (namely, loops and
coloops). Such edges are “degenerate” in some sense, and the expression on the
right-hand side is usually simpler than in the general case, with fewer terms.
•The number of terms in each of these linear expressions is fixed.
•The polynomial is well defined, in that the order in which local operations are
used when evaluating the polynomial using (1) does not matter [107].
Many other graph polynomials also satisfy reduction relations of this type. Ex-
amples include the independent set, stability, clique and matching polynomials, the
Oxley-Whittle polynomial for graphic 2-polymatroids [93, 94], the Borzacchini-Pulito
polynomial [11], and the rich class of Tutte polynomials of Hopf algebras [66].3One
of the earliest studies of graph polynomials defined by a variety of reduction relations
beyond deletion-contraction is due to Zykov [120]; this has roots in his earliest work
in graph theory [119].
Godlin, Katz and Makowsky have given a general definition of reduction relations
of roughly the above type, in the context of the logical theory of graph polynomials
[48]. Paraphrasing, they showed that every graph polynomial with such a recursive
definition in SOL can be expressed as a SOL-definable sum over subsets. This links,
in one direction, two of the main ways of defining graph polynomials: reduction
relations, and sums over subsets (“subset expansions” in their terminology). They
ask whether the link goes the other way too: does every graph polynomial expressible
as a SOL-definable sum over subsets satisfy a reduction relation of their general SOL-
based form?
Some graph polynomials do not seem to have a natural reduction relation within
the class of objects over which they are initially defined. But it often happens that,
even in such cases, there is a wider class of objects to which the graph polynomial
can be generalised and which supports a reduction relation of this type. We illustrate
this with some examples.
5.1 Counting edge-colourings
For any graph G, define P′(G;q) to be the number of q-edge-colourings of G(mirroring
the standard use of χ′(G) and χ(G) to denote the chromatic index4and chromatic
3Some notable graph polynomials are simple transformations or partial evaluations of the Tutte
polynomial, so they satisfy reduction relations of this type because the Tutte polynomial does. These
include the chromatic, flow, reliability, Martin and Jones polynomials, the Ising and Potts model
partition functions, the Whitney rank generating function and the coboundary polynomial.
4i.e., the minimum qsuch that Gis q-edge-colourable
19

number, respectively). It is well known that edge-colourings of a graph Gcorrespond
to vertex-colourings of the line graph L(G) of G(see, e.g., [43]). So we have
P′(G;q) = P(L(G); q),(5)
which makes plain that P′(G;q) a polynomial.
The most natural way to get a reduction relation for P′(G;q) is to use (5) and
invoke the deletion-contraction relation for chromatic polynomials:5
P(L(G); q) = qm,if L(G) has no edges;
P(L(G)\e;q)−P(L(G)/e;q),otherwise.
But this relation holds in the class of all graphs since, in general, L(G)\eand L(G)/e
are not line graphs.
The base cases for this reduction relation are graphs with no edges. Such a graph
is a line graph of a disjoint union of a matching and a set of isolated vertices.
5.2 The symmetric Ashkin-Teller model
In §3.2 we saw that the symmetric Ashkin-Teller model partition function is not a
specialisation of the Tutte polynomial. It therefore does not obey the kind of deletion-
contraction relation characteristic of evaluations of the Tutte polynomial.
However, it turns out that it does satisfy such a relation in the wider class of
binary functions. This follows from the following result in [34] which shows that it is
a specialisation of a suitable λ-Tutte-Whitney function.
Theorem 3 ([34]) For each Kand Kστ there exists λsuch that the partition func-
tion Z(G;K, Kστ )of the symmetric Ashkin-Teller model on a graph Gcan be obtained
from the λ-Tutte-Whitney function.
Details, including expressions for λin terms of Kand Kστ , are given in [34].
Putting Theorems 1 and 3 together gives a reduction relation for the Ashkin-Teller
model partition function in the class of binary functions.
5.3 Go polynomials
None of the Go polynomials introduced in §3.4 and [31] have an obvious recurrence
relation, of the type we have been considering, on graphs. But there are recurrence
relations in a wider class of objects that generalise graphs. In [31, §3], it is found that
Go#(G;λ) and Go(G;p) satisfy a family of linear recurrence relations over a larger
class of graphs, there called L-graphs, in which graphs may have extra labels on some
of their vertices and edges that modify the conditions that a partial λ-assignment
must satisfy in order to be a legal position.
5We assume that line graphs have no loops, which is the usual practice. If, instead, we assume
that loops are created in the line graph corresponding to loops in G, then it is natural to assume
that a line graph with a loop has no edge-colouring. If we were to allow colouring of loops in
edge-colourings, then our reduction relation would not work.
20

5.4 Polynomials based on partial colourings
We now consider reduction relations for the polynomials we introduced in §3.5.
The partial chromatic polynomial does not obey a deletion-contraction relation of
the usual type, as it is not obtainable from the Tutte polynomial and in fact contains
extra information. No reduction relation for it is given explicitly in [22]. But (2)
points to a reduction relation based on labelling. A chromatically labelled graph is
a graph in which each vertex may be labelled C, indicating that it must receive a
colour, or U, indicating that it must be uncoloured; a vertex may have no label, but
it cannot have two labels. Each chromatically labelled graph is written as G(C,U )
where C, U ⊆Vand C∩U=∅. A totally chromatically labelled graph G(C,U )is a
chromatically labelled graph in which every vertex is labelled, i.e., C∪U=V.
Apartial λ-assignment of a chromatically labelled graph G(C,U )is a partial λ-
assignment fof Gsuch that C⊆dom f⊆V\U. It is a partial λ-colouring of G(C,U)
if it is also a partial λ-colouring of G.
For chromatically labelled graphs, put
PC(G(C,U );p, λ) = Pr(fis a partial λ-colouring of G(C,U ))
= Pr((fis a λ-colouring of G[dom f]) ∧(C⊆dom f⊆V\U)) .
This polynomial, in this wider class, has a simple reduction relation.
Theorem 4 For any v∈V\(C∪U),
PC(G(C,U );p, λ) = PC(G(C∪{v},U );p, λ) + PC(G(C,U ∪{v});p, λ).
Proof.
PC(G(C,U );p, λ)
= Pr((fis a λ-colouring of G[dom f]) ∧(C⊆dom f⊆V\U))
= Pr((fis a λ-colouring of G[dom f]) ∧(C⊆dom f⊆V\U)∧(v∈dom f))
+ Pr((fis a λ-colouring of G[dom f]) ∧(C⊆dom f⊆V\U)∧(v∈ dom f))
= Pr((fis a λ-colouring of G[dom f]) ∧(C∪ {v} ⊆ dom f⊆V\U))
+ Pr((fis a λ-colouring of G[dom f]) ∧(C⊆dom f⊆V\(U∪ {v})))
= PC(G(C∪{v},U);p, λ) + PC(G(C,U ∪{v});p, λ).
The reduction relation of Theorem 4 cannot be used on totally chromatically
labelled graphs, when C∪U=V. In that case, the partial chromatic polynomial is
just a scaled version of chromatic polynomial of G−U:
PC(G(V\U,U);p, λ) = (1 −λp)|U|pn−|U|P(G−U;λ).(6)
We return to this point in §6.
21

It is also possible to get a reduction relation for PC(G;p, λ) on certain vertex-
weighted graphs using the fact that the partial chromatic polynomial is a specialisa-
tion of the U-polynomial of Noble and Welsh [92] (because its equivalent polynomial
ξ(G;x, y, z) is) which has a reduction relation on those weighted graphs.
We now consider extendable colouring polynomials and show that ECλ(G) satis-
fies a reduction relation in the class of chromatically labelled graphs we introduced
above. A partial λ-assignment of G(C,U)is λ-extendable in G(C,U )if, as a partial λ-
assignment of G, it is λ-extendable. So, although a label Uon a vertex specifies that
it is uncoloured by our random λ-assignment f, the vertex is allowed to be coloured
by an extension of f.
For chromatically labelled graphs, put
EC(G(C,U );p, λ) = Pr(fis λ-extendable in G(C,U ))
= Pr ((fis λ-extendable in G)∧(C⊆dom f⊆V\U)) .
Theorem 5 For any v∈V\(C∪U),
ECλ(G(C,U );p) = λp ECλ(G(C∪{v},U );p) + (1 −λp) ECλ(G(C,U ∪{v});p).
Proof.
ECλ(G(C,U );p) = Pr(fis λ-extendable in G(C,U ))
= Pr(fis λ-extendable in G(C,U )|v∈dom f) Pr(v∈dom f) +
Pr(fis λ-extendable in G(C,U )|v∈ dom f) Pr(v∈ dom f)
=λp Pr(fis λ-extendable in G(C∪{v},U)) +
(1 −λp) Pr(fis λ-extendable in G(C,U ∪{v}))
=λp ECλ(G(C∪{v},U);p) + (1 −λp) ECλ(G(C,U ∪{v});p).
Lastly, we consider forced colouring polynomials. A partial λ-assignment of G(C,U )
forces aλ-colouring of G(C,U)if, as a partial λ-assignment of G, it forces a λ-colouring
of G. Again, a label Uon a vertex only prevents it from being coloured by fitself;
the label does not stop it from being eventually forced by f. Put
FC(G(C,U);p, λ) = Pr(fforces a λ-colouring of G(C,U ))
= Pr ((fforces a λ-colouring of G)∧(C⊆dom f⊆V\U)) .
A very similar argument to Theorem 5 shows that FCλ(G;p) satisfies a reduction
relation in the class of chromatically labelled graphs.
Theorem 6 For any v∈V\(C∪U),
FCλ(G(C,U);p) = λp FCλ(G(C∪{v},U );p) + (1 −λp) FCλ(G(C,U ∪{v});p).
22

5.5 Questions
We have now seen six examples of graph polynomials which do not seem to satisfy
a local linear reduction relation over the class of graphs but which do satisfy such
relations over some larger class.
Other examples exist. The U-polynomial of a graph, introduced by Noble and
Welsh, has a reduction relation in the larger class of vertex-weighted graphs in which
the weights are positive integers [92] (see also [91]). Krajewski, Moffatt and Tanasa
[66] used their Hopf algebra framework to show that various topological Tutte poly-
nomials without full reduction relations6can be extended to a larger class of objects
(by augmenting the embedded graphs with some extra structure) so that they do
have full reduction relations. (See especially [66, Remark 62].)
These examples raise the question of how widespread this phenomenon is. Which
graph polynomials exhibit this phenomenon? Can they be characterised in some
formal, rigorous way? For the polynomials considered in §5.3 and §5.4, suitable su-
perclasses of the class of graphs can be found by introducing new labels on vertices
and/or edges with specific technical meanings. This is likely to be a wider phe-
nomenon and may be able to be captured using the logical framework of Makowsky
and colleagues. But our first two cases, in §5.1 and §5.2, are not of this type, and it
is not clear how to include them in a general characterisation of this phenomenon.
6 Levels of recursion
For many graph polynomials, repeated application of a reduction relation leaves only
trivial graphs. For example, repeated application of deletion-contraction relations
for the chromatic or Tutte polynomials leaves null graphs. But sometimes a graph
polynomial has a reduction relation in which the base cases are themselves nontrivial
graphs and another reduction relation needs to be applied in order to reduce them to
simpler base cases; we might say that we have two “levels” of reduction relation.
Partial chromatic polynomials provide an example. The reduction relation we
gave in Theorem 4 may be used to reduce the partial chromatic polynomial to a sum
involving partial chromatic polynomials of totally chromatically labelled graphs, and
each of these polynomials can in turn be expressed in terms of a chromatic polynomial,
by (6). So we can apply the deletion-contraction relation to each of the chromatic
polynomials, thereby expressing the partial chromatic polynomial as a sum of simple
base cases — chromatic polynomials of null graphs — with two levels of reduction.
It can get worse than this! Go polynomials (§3.4) may be regarded as having three
levels of reduction. Firstly, graphs are reduced to L-graphs using [31, Cor. 6 or The-
orem 7]. Then L-graphs are reduced to ordinary graphs again using [31, Cor. 10]. Fi-
nally, these ordinary graphs are reduced to null graphs using the deletion-contraction
relation for the chromatic polynomial. The paper in fact gives a method of expressing
a Go polynomial as a large sum of chromatic polynomials.
6because the known relations did not cover all possible edge types
23

For some polynomials, the situation is less clear. For extendable colouring and
forced colouring polynomials, we gave reduction relations in Theorems 5 and 6 whose
base cases require computation of those polynomials for totally chromatically labelled
graphs. Those computations do not have obvious analogues of (6).
For extendable colouring polynomials, here is an attempt involving an addition-
identification relation on the set of vertices labelled C:
ECλ(G(C,U );p) = ECλ(G(C,U )+uv;p) + ECλ(G(C,U )/uv;p),
for any u, v ∈Csuch that uv ∈ E(G). This can be applied repeatedly until the
vertices labelled Cform a clique in G. Whenever this clique has > λ vertices, the
polynomial is identically 0 since the graph is not λ-colourable. So we end up with
a sum of polynomials ECλ(H(D,U );p) of totally chromatically labelled graphs of the
form H(D,U )where Dis a clique of size ≤λin H. Because Dis a clique, a partial
λ-assignment of H(D,U)is λ-extendable if and only if His λ-colourable. So we have
ECλ(H(D,U );p)
= Pr ((fis λ-extendable in H)∧(D⊆dom f⊆V\U)∧(fis a λ-colouring of H[D]))
= Pr ((D⊆dom f⊆V\U)∧(fis a λ-colouring of H[D])) ×
Pr ((fis λ-extendable in H)|(D⊆dom f⊆V\U)∧(fis a λ-colouring of H[D]))
=p|D|(λ)|D|·JHis λ-colourable K.
Here we have used the Iverson bracket:
JHis λ-colourable K=1,if His λ-colourable;
0,otherwise.
We seem to have gained something, computationally, by expressing it this way: we
“only” have an NP-complete quantity to evaluate, rather than a #P-hard graph
polynomial! But it is no longer a natural sum of graph polynomials, so it does not
give us another layer of reduction relations of the kind we have been considering.
We can try a similar approach with forced colouring polynomials (Theorem 6).
Again, the base cases for the first reduction relation we use are totally chromatically
labelled graphs, and again we can use addition-identification repeatedly to get a
sum over totally chromatically labelled graphs H(D,U )in which the set Dof vertices
labelled Cinduces a clique of size ≤λ. The summand for H(D,U)includes the factor
Jaλ-colouring of H[D] forces a λ-colouring of HK,
again using the Iverson bracket. This is polynomial-time computable, so we could
reasonably call it a final base case and say that forced λ-colouring polynomials have
two levels of recursion. But these base cases are much less simple than the base cases
for other recursions we have considered (e.g., null graphs, for chromatic polynomials
and (eventually) for Go polynomials). This time, the computational task for each
base case is P-complete [30].
It would be interesting to study this phenomenon of levels of recursion for graph
polynomials more systematically. Perhaps the notion can be formalised and then
related to the logical structure of the definition of the polynomial.
24

7 Graph polynomials?
We perhaps take it for granted that graph invariants giving counts, or probabilities,
of structures of interest are polynomials. This is not a necessary feature of such
invariants. In general, the λ-Whitney function of a graph [32] may have irrational
exponents, though in certain forms this can be avoided when evaluating them along
hyperbolae xy = 2rfor r∈N. (Some related polynomials in knot theory may have
negative or fractional exponents, but this is not a significant exception because in
general they can be transformed to polynomials by appropriate changes of variable.)
The choice of parameter is crucial. For example, define HomCyc(G;q) to be the
number of homomorphisms from Gonto the cycle Cq. Such homomorphisms may
be viewed as q-assignments in which adjacent vertices in Gare mapped to “colours”
(being vertices in Cq) that are adjacent in Cq; unlike normal graph colouring (when
the homomorphism is to Kq), the two distinct colours used on the endpoints of an
edge cannot be completely arbitrary but are constrained to be neighbouring colours
(vertices) in Cq. This is not, in general, a polynomial in q. One way to see this is to
note that HomCyc(K3;q) = 0 for all even qbut is not identically 0. In other words,
in the terminology of de la Harpe and Jaeger [19], the sequence (Cn:n∈N) is not a
strongly polynomial sequence of graphs. See [50] for a study of the deep question of
which sequences of graphs, treated as targets of homomorphisms, give rise to graph
polynomials that count homomorphisms.
Graph colouring requires the colour classes to induce null graphs, where the
chromons each consist of a single vertex. There has been a lot of work on gener-
alised colourings where the chromons are less severely restricted. One of the simplest
relaxations is to bound the sizes of the chromons. Define mc(G;s) to be the number
of 2-assignments of Gin which every chromon has size ≤s. We have
mc(G; 0) = 1,if n= 0,
0,if n≥1;
mc(G; 1) = 2k(G),if Gis bipartite,
0,otherwise;
mc(G;s) = 2n∀s≥n.
So mc(G;s) cannot be a polynomial in s.
Define emb(G;g) to be the number of orientable 2-cell combinatorial embeddings
of Gof genus g, where embeddings are given by rotation schemes. This cannot be a
polynomial in gbecause it is positive when glies between the genus and maximum
genus, inclusive, of G, but is zero for all gabove the maximum genus.
For graph invariants that are polynomials, it is natural to hope that the theory of
polynomials may shed light on the graph polynomials and, through them, on graphs
themselves. After all, polynomials have a rich mathematical theory that has been
built up over centuries.
For example, Birkhoff’s work on the chromatic polynomial, beginning with [8], was
motivated by the thought that its properties, as a polynomial, might help prove the
25

Four-Colour Conjecture (as it then was). According to Morse [90, p. 386], “Birkhoff
hoped that the theory of chromatic polynomials could be so developed that methods
of analytic function theory could be applied.”
With this motivation, we can ask of any graph polynomial, which aspects of the
theory of polynomials correspond to properties of the underlying graph?
It is common for graph polynomials to be multiplicative over components or even
over blocks (with multiplicativity over blocks being typical for polynomials that de-
pend only on the cycle matroid of the graph, such as the Tutte polynomial). This
is to be expected for polynomials that count things or give probabilities, since the
lack of interaction between separate components or blocks typically means we can
treat them as contributing independently to counts or probabilities. Does such a
relationship work both ways? In other words, does multiplicativity only occur over
components/blocks? What, in the graph, is represented by the polynomial’s proper
factors? How can we characterise the structure of graphs whose polynomial is irre-
ducible?
In the case of the Tutte polynomial, it was shown by Merino, de Mier and Noy that
the Tutte polynomial of a matroid is irreducible if and only if the matroid is connected
[80]. So the factors of the polynomial correspond exactly to the components of the
matroid, which means that, for graphs, they correspond to blocks.
But the situation is not so straightforward for many other graph polynomials,
even for those that are specialisations of the Tutte polynomial.
For the chromatic polynomial, it was known to Whitney [116, §14] that the chro-
matic polynomial factorises when the graph is clique-separable7in the sense that it
has a separating clique, where a separating clique is a clique whose removal increases
the number of components of the graph. If Gis formed from overlap of H1and H2
in a separating r-clique, then Whitney showed that
P(G;x) = P(H1;x)P(H2;x)
P(Kr;x).
Somewhat surprisingly, such chromatic factorisations can occur in other cases, too: in
[87, 88], examples are given of chromatic factorisations of graphs that are strongly non-
clique-separable in that they are not chromatically equivalent to a clique-separable
graph. Some studies of this kind have since been done for other polynomials including
the reliability polynomial [86] and the stability polynomial [81].
Another fundamental topic in the mathematical theory of polynomials is Galois
theory. So it is natural to ask about the relationship between the structure of a graph
and the Galois group of a graph polynomial derived from it. An initial investigation of
this topic for the chromatic polynomial, including computational results, is reported
in [85].
The most fundamental property in any mathematical system is identity: when are
two objects considered the same? For a graph polynomial, this is when two graphs are
equivalent in the sense that they have the same polynomial. This leads to the notion
7but keep in mind that this term also has a completely different meaning [47]
26

of certificates of equivalence, which can also be adapted to chromatic factorisation
and which we discuss in the next section.
8 Certificates
The notion of a certificate to explain chromatic factorisation and chromatic equiva-
lence was introduced by Morgan and Farr [87, 88] and developed further in [83, 84, 12].
The idea has since been extended to other graph polynomials including the stability
polynomial [81], reliability polynomial [86] and Tutte polynomial [81]. This use of
the term “certificate” is inspired by its use in complexity theory, e.g., in defining NP,
but we are using the term in a much more specific sense.
Informally, a certificate is a sequence of expressions E0, E1, E2, . . . , Ekin graphs
where each expression Ei,i > 0, can be obtained from its predecessor Ei−1by applying
a relation satisfied by the graph polynomial in question (e.g., a deletion-contraction
relation, or multiplicativity for disjoint unions). In these expressions, a graph can be
regarded as representing its corresponding polynomial. Replacing each graph by its
polynomial, then simplifying the entire expression, gives a polynomial that can be
thought of as the graph polynomial for that expression.
An example is given in Figure 4. In this certificate, we use the deletion-contraction
relation for the Tutte polynomial in the form
[T1]: G−→ G\e+G/e,
with each graph standing for its Tutte polynomial.
A simple rearrangement of T1 and renaming of the graphs used in the terms gives
us
[T2]: G−→ (G+uv)−G/uv.
Figure 4: Certificate of Tutte Equivalence. Graph Gis Tutte Equivalent to Graph
H.
27

The resulting certificate has the form:
G=G\e+G/e (Applying T1)
= (G\e+uv)−G\e/uv +G/e (Applying T2)
=G\e+uv (Algebraic Step (cancellation)) (7)
where the graph G\e+uv ∼
=H. If we replace each graph by its Tutte polynomial,
each expression in the sequence is equal to the Tutte polynomial of G, and hence we
have a certificate that shows that graphs Gand Hare Tutte equivalent.
Our work so far has focused on certificates of equivalence and factorisation. We
expect that certificates could provide a graph-theoretic approach to studying other
algebraic properties of graph polynomials.
In [83], Morgan introduced the concept of a schema or template for certificates
of factorisation and equivalence. A schema specifies the structure of a certificate,
including the relation to be applied at each step, but without filling in the actual
graphs. So, instead of actual graphs (as in Figure 4), we just have symbols for
them. In fact, as written — with symbols G,G\e, etc. — (7) is really a schema for
certificates, and the certificate in Figure 4 is one particular certificate that belongs
to this schema.
In [87, 88], it was shown that every graph in a particular infinite family of strongly
non-clique-separable graphs has a chromatic factorisation. Each certificate of factori-
sation in this family used the same sequence of certificate steps, so the entire infinite
family of certificates could be described by a single schema.
If two graphs have the same multiset of blocks, then they are Tutte equivalent,
since the Tutte polynomial is multiplicative over blocks. We can capture this multi-
plicativity in certificate steps that allow a graph to be replaced by a formal product
of its blocks and vice versa.
[T3]: G−→ B1B2· · · Bkwhere the Biare the blocks of G,
[T4]: B1B2· · · Bk−→ Gwhere Gis a graph with blocks Bi, 1 ≤i≤k,
This enables us to write the following simple schema for certificates of Tutte equiva-
lence for pairs of graphs with the same blocks.
G=
k
Y
i=1
Bi(Applying T3)
=H(Applying T4). (8)
Effectively, this schema first ‘unglues’ blocks then ’glues’ them back together to pro-
duce graph H. This certificate schema works for all pairs of graphs that have the
same blocks and may be regarded as a representation of the set of all such pairs.
Schemas 1 and 2 in [12] give two of the shortest certificates for pairs of chromati-
cally equivalent graphs, Gand H. In both these schemas, graph Hcan be obtained
by removing an edge from graph Gand then adding a different edge.
28

Schema 2 relates pairs of chromatically equivalent graphs Gand Hwith G\e∼
=
H\fand G/e ∼
=H/f , where e∈E(G) and f∈E(H). Applying two deletion-
contraction steps, we have:
G=G\e−G/e
= (G\e) + f(9)
where f∈ E(G) and G\e+f∼
=H. The second step, (9), is obtained by rearranging
the usual deletion-contraction relation.
Schema 1 is similar to Schema 2, but uses the addition-identification relation.
Here (G+e)\f∼
=H. Applying two addition-identification steps, we have:
G=G+e+G/e
= (G+e)\f(10)
where e∈ E(G) and f∈E(G).
More sophisticated certificates of equivalence are available. In [12], shortest cer-
tificates of chromatic equivalence are given for all pairs of chromatically equivalent
graphs of order at most 7. These corresponded to 15 different schemas. It should be
noted that a shortest certificate of equivalence may not be unique. In [84], infinitely
many pairs of chromatically equivalent non-isomorphic graphs are constructed along
with their certificates of equivalence.
The length of certificates has implications for the complexity of testing equivalence
with respect to these polynomials (chromatic equivalence, Tutte equivalence, etc.).
A short certificate of equivalence, once obtained, gives a means of verifying that two
graphs have the same polynomial without computing the polynomial. For example,
if the length of certificates of chromatic equivalence is polynomially bounded, then
the problem of testing chromatic equivalence belongs to NP [12]. But at present we
only have very loose, exponential upper bounds on certificate length. We discuss
some implications of certificate length for the computational complexity of chromatic
equivalence, factorisation and uniqueness in [12] and Tutte equivalence in [82].
Appropriate versions of certificates of equivalence and factorisation should be ap-
plicable to many other graph polynomials. We would like to see a rigorous theory
of certificates of (at least) equivalence and factorisation for a broad class of graph
polynomials with reduction relations. The commutativity of the operations of delet-
ing/contracting different edges in a graph may be regarded as an instance of the
Church-Rosser property from the theory of rewriting systems, as observed by Yetter
[118] and Makowsky [70]; see also [7, result 9m, p. 72]. It seems to us that the theory
of rewriting systems could shed more light on the kind of certificates we have consid-
ered here.
Acknowledgements
We thank Andrew Goodall, J´anos Makowsky, Steven Noble and the referee for
their comments. Through this Festschrift contribution, it is a pleasure to acknowl-
edge J´anos’s far-reaching contributions to the study of graph polynomials, through
29

his mathematics and also through his generosity as a colleague, in sharing ideas,
organising meetings and supporting the work of others.
References
[1] J.L. Arocha and B. Llano, Mean value for the matching and dominating poly-
nomial, Discussiones Mathematicae Graph Theory 20 (2000) 57–69.
[2] J. Ashkin and E. Teller, Statistics of two-dimensional lattices with four compo-
nents, Phys. Rev. 64 (1943) 178–184.
[3] I. Averbouch, B. Godlin and J.A. Makowsky, A most general edge elimination
polynomial, in: H. Broersma, T. Erlebach, T. Friedetzky and D. Paulusma
(eds.), Graph-Theoretic Concepts In Computer Science: 34th Internat. Work-
shop (WG 2008) (Durham, UK, June/July 2008), Lecture Notes in Comput.
Sci. 5344, Springer, Berlin, 2008, pp. 31–42.
[4] I. Averbouch, B. Godlin and J.A. Makowsky, An extension of the bivariate
chromatic polynomial, European J. Combin. 31 (1) (2010) 1–17.
[5] L. Beaudin, J. Ellis-Monaghan, G. Pangborn and R. Shrock, A little statistical
mechanics for the graph theorist, Discrete Math. 310 (2010) 2037–2053.
[6] D. B. Benson, Life in the game of Go, Information Sciences 17 (1976) 17–29.
[7] N. L. Biggs, Algebraic Graph Theory (2nd edn.), Cambridge University Press,
1993.
[8] G. D. Birkhoff, A determinant formula for the number of ways of coloring a
map, Ann. of Math. 14 (1912–1913) 42–46.
[9] M. Bl¨aser, H. Dell and J.A. Makowsky, Complexity of the Bollob´as-Riordan
polynomial. Exceptional points and uniform reductions, Theory Comput. Syst.
46 (4) (2010) 690–706.
[10] A. Bohn, Chromatic roots as algebraic integers, 24th Internat. Conf. on Formal
Power Series and Algebraic Combinatorics (FPSAC 2012) (Nagoya, Japan, 30
July – 3 Aug 2012), pp. 539–550.
[11] L. Borzacchini and C. Pulito, On subgraph enumerating polynomials and Tutte
polynomials, Boll. Un. Mat. Ital. B (6) 1(1982) 589–597.
[12] Z. Bukovac, G. Farr and K. Morgan, Short certificates for chromatic equivalence,
Journal of Graph Algorithms and Applications 23 (2) (2019) 227–269.
[13] P.J. Cameron, Cycle index, weight enumerator, and Tutte polynomial, Electron.
J. Combin. 9(2002) #N2 (10pp).
30

[14] P.J. Cameron, Orbit counting and the Tutte polynomial, n: G. R. Grimmett
and C. J. H. McDiarmid (eds.), Combinatorics, Complexity and Chance: A
Tribute to Dominic Welsh, Oxford University Press, 2007, pp. 1–10.
[15] P.J. Cameron, B. Jackson and J.D. Rudd, Orbit-counting polynomials for
graphs and codes, Discrete Math. 308 (5–6) (2008) 920–930.
[16] P.J. Cameron and K. Morgan, Algebraic properties of chromatic roots, Elec-
tronic J. Combinatorics 24 (2017) #P1.21.
[17] S. Chmutov, Topological extensions of the Tutte polynomial, in: J. Ellis-
Monaghan and I. Moffatt (eds.), Handbook on the Tutte Polynomial and Related
Topics, Chapman and Hall/CRC Press, 2022, Chapter 27, pp. 497–513
[18] B. Courcelle, J.A. Makowsky and U. Rotics, On the fixed parameter complexity
of graph enumeration problems definable in monadic second-order logic, Dis-
crete Appl. Math. 108 (1–2) (2001) 23–52.
[19] P. de la Harpe and F. Jaeger, Chromatic invariants for finite graphs: theme
and polynomial variations, Linear Algebra Appl. 226–228 (1995) 687–722.
[20] D. Delbourgo and K. Morgan, Algebraic invariants arising from chromatic poly-
nomials of theta graphs, Australasian Journal of Combinatorics 59 (2) (2014)
293–310.
[21] A. de Mier Vinu’e, Graphs and Matroids Determined by their Tutte Polynomi-
als, PhD thesis, Universitat Polit‘ecnica de Catalunya, 2003.
[22] K. Dohmen, A. P¨onitz and P. Tittmann, A new two-variable generalization
of the chromatic polynomial, Discrete Mathematics and Theoretical Computer
Science 6(2003) 069–090.
[23] J. Ellis-Monaghan, A. Goodall, J. A. Makowsky and I. Moffatt, Graph Polyno-
mials: Towards a Comparative Theory (Dagstuhl Seminar 16241), in: Dagstuhl
Reports (Schloss Dagstuhl - Leibniz-Zentrum f¨ur Informatik) 6(6) (2016) 26–
48. https://doi.org/10.4230/DagRep.6.6.26
[24] J. Ellis-Monaghan, A. Goodall, I. Moffatt and K. Morgan, Comparative Theory
for Graph Polynomials (Dagstuhl Seminar 19401), in: Dagstuhl Reports (Schloss
Dagstuhl - Leibniz-Zentrum f¨ur Informatik) 9(9) (2019) 135–155. https://
doi.org/10.4230/DagRep.9.9.135
[25] J. Ellis-Monaghan and I. Moffatt, Graphs on Surfaces: Dualities, Polynomials,
and Knots, Springer, 2013.
[26] J. Ellis-Monaghan and I. Moffatt (eds.), Handbook on the Tutte Polynomial and
Related Topics, Chapman and Hall/CRC Press, 2022.
31

[27] J. W. Essam, Graph theory and statistical physics, Discrete Math. 1(1971)
83–112.
[28] G. E. Farr, A correlation inequality involving stable set and chromatic polyno-
mials, J. Combin. Theory (Ser. B) 58 (1993) 14–21.
[29] G. E. Farr, A generalization of the Whitney rank generating function, Math.
Proc. Camb. Phil. Soc. 113 (1993) 267–280.
[30] G. E. Farr, On problems with short certificates, Acta Informatica 31 (1994)
479–502.
[31] G. E. Farr, The Go polynomials of a graph, Theoretical Computer Science 306
(2003) 1–18.
[32] G. E. Farr, Some results on generalised Whitney functions, Adv. in Appl. Math.
32 (2004) 239–262.
[33] G. E. Farr, Tutte–Whitney polynomials: some history and generalizations, in:
G. R. Grimmett and C. J. H. McDiarmid (eds.), Combinatorics, Complexity
and Chance: A Tribute to Dominic Welsh, Oxford University Press, 2007, pp.
28–52.
[34] G. E. Farr, On the Ashkin–Teller model and Tutte–Whitney functions, Combin.
Probab. Comput. 16 (2007) 251–260.
[35] G. E. Farr, Transforms and minors for binary functions, Ann. Combin. 17
(2013) 477–493.
[36] G. E. Farr, The probabilistic method meets Go, Journal of the Korean Mathe-
matical Society 54 (2017) 1121–1148.
[37] G. E. Farr, Minors for alternating dimaps, Q. J. Math. 69 (2018) 285–320.
[38] G. E. Farr, Binary functions, degeneracy, and alternating dimaps, Discrete
Mathematics 342 (5) (2019) 1510–1519.
[39] G. E. Farr, Using Go in teaching the theory of computation, SIGACT News 50
(1) (March 2019) 65–78.
[40] G. E. Farr, The history of Tutte–Whitney polynomials, in: J. Ellis-Monaghan
and I. Moffatt (eds.), Handbook on the Tutte Polynomial and Related Topics,
Chapman and Hall/CRC Press, 2022, Chapter 34, pp. 623–668.
[41] G. E. Farr, The forced colouring function of a graph, in preparation.
[42] G. E. Farr and J. Schmidt, On the number of Go positions on lattice graphs,
Information Processing Letters 105 (2008) 124–130.
32

[43] S. Fiorini and R.J. Wilson, Edge-Colourings of Graphs, Research Notes in Math-
ematics 16, Pitman, 1977.
[44] E. J. Farrell, An introduction to matching polynomials, J. Combin. Theory Ser.
B27 (1979) 75–86.
[45] E. J. Farrell, On a class of polynomials associated with the cliques in a graph
and its applications, Internat. J. Math. Math. Sci. 12 (1989) 77–84.
[46] C. M. Fortuin and P. W. Kasteleyn, On the random cluster model. I. Introduc-
tion and relation to other models, Physica 57 (1972) 536–564.
[47] F. Gavril, Algorithms on clique separable graphs, Discrete Math. 19 (1977)
159–165.
[48] B. Godlin, E. Katz and J.A. Makowsky, Graph polynomials: from recursive
definitions to subset expansion formulas, J. Logic Comput. 22 (2) (2012) 237–
265.
[49] C.D. Godsil and I. Gutman, On the theory of the matching polynomial, J.
Graph Theory 5(1981) 137–144..
[50] A.J. Goodall, J. Neˇsetˇril and P. Ossona de Mendez, Strongly polynomial se-
quences as interpretations, J. Appl. Logic 18 (2016) 129–149.
[51] R. Greenlaw, H. J. Hoover and W. L. Ruzzo, Limits to Parallel Computation:
P-Completeness Theory, Oxford University Press, 1995.
[52] I. Gutman and F. Harary, Generalizations of the matching polynomial, Utilitas
Mathematica 24 (1983) 97–106.
[53] F. Harary and E. Palmer, Graphical Enumeration, Academic Press, New York,
1973.
[54] O. J. Heilmann and E. H. Lieb, Monomers and dimers, Phys. Rev. Lett. 24 (25)
(1970) 1412–1414.
[55] T. Helgason, Aspects of the theory of hypermatroids, in: C. Berge and D. K.
Ray-Chaudhuri (eds.), Hypergraph Seminar, Lecture Notes in Mathematics 411,
Springer, Berlin, 1974, pp. 191–213.
[56] S. Huggett, The Tutte polynomial and knot theory, in: J. Ellis-Monaghan and I.
Moffatt (eds.), Handbook on the Tutte Polynomial and Related Topics, Chapman
and Hall/CRC Press, 2022, Chapter 18, pp. 352–367.
[57] T. Hunt, Milton Keynes Go Board, British Go Association, 30 April 2013.
https://www.britgo.org/clubs/mk/mkboard.html
33

[58] E. Ising, Beitrag zur Theorie des Ferromagnetismus, Z. Phys. 31 (1925) 253–
258.
[59] K. Iwamoto, Go for Beginners, Ishi Press, 1972, and Penguin Books, 1976.
[60] T. Kotek, Definability of Combinatorial Functions, PhD thesis, Computer Sci-
ence Department, Technion, 2012.
[61] T.A. Kotek and J.A. Makowsky, The exact complexity of the Tutte polynomial,
in: J. Ellis-Monaghan and I. Moffatt (eds.), Handbook on the Tutte Polynomial
and Related Topics, Chapman and Hall/CRC Press, 2022, Chapter 9, pp. 175–
193.
[62] T.A. Kotek, J.A. Makowsky and E.V. Ravve, A computational framework for
the study of partition functions and graph polynomials, in: R. Downey, J. Bren-
dle, R. Goldblatt and B. Kim (eds.), Proc. 12th Asian Logic Conf. (Wellington,
15–20 December 2011), World Scientific, Hackensack, NJ, 2013, pp. 210–230.
[63] T. Kotek, J.A. Makowsky and E.V. Ravve, On sequences of polynomials arising
from graph invariants, European J. Combin. 67 (2018) 181–198.
[64] T. Kotek, J.A. Makowsky and B. Zilber, On counting generalized colorings,
in: M. Kaminski and S. Martini (eds.), Computer Science Logic: Proc. 22nd
Internat. Workshop (CSL 2008), 17th Ann. Conf. of the EACSL (Bertinoro,
16–19 September 2008), Lecture Notes in Computer Science 5213, Springer,
Berlin, 2008, pp. 339–353.
[65] M. Lotz and J.A. Makowsky, On the algebraic complexity of some families of
coloured Tutte polynomials, Adv. in Appl. Math. 32 (1–2) (2004) 327–349.
[66] T. Krajewski, I. Moffatt and A. Tanasa, Hopf algebras and Tutte polynomials,
Adv. in Appl. Math. 95 (2018) 271–330.
[67] J.A. Makowsky, Algorithmic uses of the Feferman–Vaught Theorem, Annals of
Pure and Applied Logic 126 (2004) 159–213.
[68] J.A. Makowsky, Coloured Tutte polynomials and Kauffman brackets for graphs
of bounded tree width, Discrete Appl. Math. 145 (2) (2005) 276–290.
[69] J.A. Makowsky, From a zoo to a zoology: descriptive complexity for graph poly-
nomials, in: A. Beckmann, U. Berger, B. L¨owe, and J.V. Tucker (eds.), Logical
Approaches to Computational Barriers, Second Conference on Computability
in Europe, CiE 2006 (Swansea, UK, July 2006), Lecture Notes in Computer
Science 3988, Springer, 2006, pp. 330–341.
[70] J.A. Makowsky, From a zoo to a zoology: towards a general theory of graph
polynomials, Theory of Computing Systems 43 (2008) 542–562.
34

[71] J.A. Makowsky, The graph polynomials project in Haifa, https://janos.cs.
technion.ac.il/RESEARCH/gp-homepage.html [accessed 23 February 2024].
[72] J.A. Makowsky, How I got to like graph polynomials, arXiv preprint, 6 Septem-
ber 2023, http://arxiv.org/abs/2309.02933v1.
[73] J.A. Makowsky, Meta-theorems for graph polynomials, to appear.
[74] J.A. Makowsky and J.P. Mari˜no, Farrell polynomials on graphs of bounded tree
width, Adv. in Appl. Math. 30 (1–2) (2003) 160–176.
[75] J.A. Makowsky and K. Meer, On the complexity of combinatorial and metafi-
nite generating functions of graph properties in the computational model of
Blum, Shub and Smale, in: P.G. Clote and H. Schwichtenberg (eds.), Computer
Science Logic: Proc. 14th Internat. Workshop (CSL 2000), Ann. Conf. of the
EACSL (Fischbachau, 21–26 August 2000), Lecture Notes in Computer Science
1862, Springer-Verlag, Berlin, 2000, pp. 399–410.
[76] J.A. Makowsky and E.V. Ravve, Semantic equivalence of graph polynomials
definable in second order logic, in: Jouko V¨a¨an¨anen, ˚
A. Hirvonen and R. de
Queiroz (eds.), Logic, Language, Information, and Computation: Proc. 23rd
Internat. Workshop (WoLLIC 2016) (Benem´erita Universidad Aut´onoma de
Puebla, Puebla, 16–19 August 2016), Lecture Notes in Comput. Sci. 9803,
Springer-Verlag, Berlin, 2016, pp. 279–296.
[77] J.A. Makowsky, E.V. Ravve and N.K. Blanchard, On the location of roots of
graph polynomials, European J. Combin. 41 (2014) 1–19.
[78] J.A. Makowsky, E.V. Ravve and T. Kotek, A logician’s view of graph polyno-
mials, Annals of Pure and Applied Logic 170 (2019) 1030–1069.
[79] P. Martin, Anneau de Tutte–Grothendieck associ´e aux d´enombrements eul´eriens
dans les graphes 4-r´eguliers planaires, Colloque sur la Th´eorie des Graphes
(Brussels, 1973), Cahiers Centre ´
Etudes Recherche Op´er. 15 (1973) 343–349.
[80] C. Merino, A. de Mier and M. Noy, Irreducibility of the Tutte polynomial of a
connected matroid, Journal of Combinatorial Theory 83 (2001) 298–304.
[81] R Mo, G Farr and K Morgan, Certificates for properties of stability polynomials
of graphs, Electronic Journal of Combinatorics 21 (2014), #P1.66 (25pp).
[82] R. Mo, K. Morgan and G. Farr, Certificates for Tutte equivalence, to be sub-
mitted.
[83] K. Morgan, Algebraic aspects of the chromatic polynomial, PhD thesis, Monash
University, 2010. https://doi.org/10.4225/03/587852a8b487b
35

[84] K. Morgan, Pairs of chromatically equivalent graphs, Graphs and Combinatorics
27 (4) (2010) 547–556.
[85] K. Morgan, Galois groups of chromatic polynomials, LMS J. Comput. Math.
15 (2012) 281–307.
[86] K. Morgan and R. Chen, An infinite family of 2-connected graphs that have
reliability factorisations, Discrete Applied Mathematics 218 (2017) 123–127.
[87] K. Morgan and G. Farr, Certificates of factorisation for chromatic polynomials,
Electron. J. Combin. 16 (2009) #R74 (29pp).
[88] K. Morgan and G. Farr, Certificates of factorisation for a class of triangle-free
graphs, Electron. J. Combin. 16 (2009) #R75 (14pp).
[89] K. Morgan and G. Farr, Non-bipartite chromatic factors, Discrete Mathematics
312 (2012) 1166–1170.
[90] M. Morse, George David Birkhoff and his mathematical work, Bull. Amer.
Math. Soc. 52 (1946) 357–391.
[91] S. D. Noble, The U,V, and Wpolynomials, in: J. Ellis-Monaghan and I.
Moffatt (eds.), Handbook on the Tutte Polynomial and Related Topics, Chapman
and Hall/CRC Press, 2022, Chapter 26, pp. 7733–496.
[92] S. D. Noble and D. J. A. Welsh, A weighted graph polynomial from chromatic
invariants of knots, Ann. Inst. Fourier (Grenoble) 49 (3) (1999) 1057–1087.
[93] J. G. Oxley and G. P. Whittle, Tutte invariants for 2-polymatroids, in: N.
Robertson and P. D. Seymour (eds.), Graph Structure Theory (Seattle, 1991),
Contemp. Math. 147, Amer. Math. Soc., Providence, RI, 1993, pp. 9–19.
[94] J. Oxley and G. Whittle, A characterization of Tutte invariants of 2-
polymatroids, J. Combin. Theory Ser. B 59 (1993) 210–244.
[95] G. P´olya, Kombinatorische Anzahlbestimmungen f¨ur Gruppen, Graphen und
chemische Verbindungen, Acta Math. 68 (1937) 145–254.
[96] R. B. Potts, Some generalized order-disorder transformations, Proc. Camb. Phil.
Soc. 48 (1952) 106–109.
[97] H. Redfield, The theory of group-reduced distributions, Amer. J. Math. 49
(1927) 433–455.
[98] H. Segerman, Diamond Go, http://www.segerman.org/diamond/.
[99] Sensei’s Library, Tromp-Taylor Rules, https://senseis.xmp.net/
?TrompTaylorRules [accessed 19 February 2024].
36

[100] R. P. Stanley, Acyclic orientations of graphs, Discrete Math. 5(1973) 171–178.
[101] E. Thorp and W.E. Walden, A partial analysis of Go, Computer J. 7(3) (1964)
203–207.
[102] E. Thorp and W.E. Walden, A computer assisted study of Go on M×Nboards,
Information Sciences 4(1) (1972) 1–33.
[103] P. Tittmann, Graph Polynomials: The Eternal Book, 14 February
2024. https://www.researchgate.net/publication/377572474_Graph_
Polynomials_The_Eternal_Book
[104] M. Trinks, The covered components polynomial: a new representation of the
edge elimination polynomial, Electron. J. Combin. 19 (2012) #P50 (31pp).
[105] J. Tromp and W. Taylor, The game of Go, https://tromp.github.io/go.
html [accessed 19 February 2024].
[106] W. T. Tutte, A ring in graph theory, Proc. Camb. Phil. Soc. 43 (1947) 26–40.
[107] W. T. Tutte, A contribution to the theory of chromatic polynomials, Canadian
J. Math. 6(1954) 80–91.
[108] W. T. Tutte, On dichromatic polynomials, J. Combin. Theory 2(1967) 301–
320.
[109] W. T. Tutte, Codichromatic graphs, J. Combin. Theory Ser. B 16 (1974) 168–
174.
[110] W. T. Tutte, Graph Theory as I Have Known It, Oxford, 1998.
[111] W. T. Tutte, Graph-polynomials, Adv. in Appl. Math. 32 (2004) 5–9.
[112] R. Van Slyke and H. Frank, Network reliability analysis. I., Networks 1
(1971/72), 279–290.
[113] D. J. A. Welsh, Matroid Theory, London Math. Soc. Monograph No. 8, Aca-
demic Press, London, 1976.
[114] D. J. A. Welsh, Complexity: Knots, Colourings and Counting, London Math.
Soc. Lecture Note Series 186, Cambridge University Press, 1993.
[115] H. Whitney, The Coloring of Graphs, PhD thesis, Harvard University, 1932.
[116] H. Whitney, The coloring of graphs, Ann. of Math. (2) 33 (1932) 688–718.
[117] H. Whitney, A set of topological invariants for graphs, Amer. J. Math. 55 (1933)
231–235.
37

[118] D. N. Yetter, On graph invariants given by linear recurrence relations, J. Com-
bin. Theory (Ser. B) 48 (1990) 6–18.
[119] A. A. Zykov, On some properties of linear complexes (in Russian), Math. Sbornik
24 (1949) 163–188. English translation: Amer. Math. Soc. Transl. 79 (1952).
[120] A. A. Zykov, Recursively calculable functions of graphs, in: Theory of Graphs
and its Applications (Proc. Sympos. Smolenice, 1963), Publ. House Czechoslo-
vak Acad. Sci., Prague, 1964, pp. 99–105.
38