On the decomposition of connected graphs into their biconnected components
ABSTRACT We give a recursion formula to generate all equivalence classes of biconnected graphs with coefficients given by the inverses of the orders of their groups of automorphisms. We give a linear map to produce a connected graph with say, u, biconnected components from one with u1 biconnected components. We use such map to extend the aforesaid result to connected or 2edge connected graphs. The underlying algorithms are amenable to computer implementation.

Article: Combinatorics of 1particle irreducible npoint functions via coalgebra in quantum field theory
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ABSTRACT: We give a coalgebra structure on 1vertex irreducible graphs which is that of a cocommutative coassociative graded connected coalgebra. We generalize the coproduct to the algebraic representation of graphs so as to express a bare 1particle irreducible npoint function in terms of its loop order contributions. The algebraic representation is so that graphs can be evaluated as Feynman graphs.Journal of Mathematical Physics 01/2010; 51(8). · 1.18 Impact Factor
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arXiv:1001.3163v2 [math.CO] 13 Jun 2010
On the decomposition of connected
graphs into their biconnected
components
ˆAngela Mestre∗
Institut de Min´ eralogie et de Physique des Milieux Condens´ es,
Universit´ e Pierre et Marie Curie, Campus Boucicaut,
140 rue de Lourmel, F75015 Paris
France
June 15, 2010
Abstract
We give a recursion formula to generate all equivalence classes of bi
connected graphs with coefficients given by the inverses of the orders of
their groups of automorphisms. We give a linear map to produce a con
nected graph with say, µ, biconnected components from one with µ − 1
biconnected components. We use such map to extend the aforesaid result
to connected or 2edge connected graphs. The underlying algorithms are
amenable to computer implementation.
1Introduction
As pointed out in [12], generating graphs may be useful for numerous reasons.
These include giving more insight into enumerative problems or the study of
some properties of graphs.Problems of graph generation may also suggest
conjectures or point out counterexamples.
In particular, the problem of generating graphs taking into account their
symmetries was considered as early as the 19thcentury [5]. Sometimes such
problem is so that graphs are weighted by scalars given by the inverses of the
orders of their groups of automorphisms. One instance is [11] (page 209). Many
∗Present address: Centro de Estruturas Lineares e Combinat´ orias, Complexo Interdisci
plinar da Universidade de Lisboa, Av. Prof. Gama Pinto 2, 1649003 Lisboa Portugal; email:
mestre@alf1.cii.fc.ul.pt.
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other examples may be found in mathematical physics (see for instance [4] and
references therein).In this context, the problem is traditionally dealt with
via generating functions and functional derivatives. However, any method to
straightforwardly manipulate graphs may actually be used. In particular, the
main results of [9] and [10] are recursion formulas to generate all equivalence
classes of trees and connected graphs (with multiple edges and loops allowed),
respectively, via Hopf algebra. The key feature is that the sum of the coefficients
of all graphs in the same equivalence class is given by the inverse of the order
of their automorphism group.
Furthermore, in [7] the algorithm underlying the main result of [10] was
translated to the language of graph theory. To this end, basic graph transfor
mations whose action mirrors that of the Hopf algebra structures considered in
the latter paper, were given. The result was also extended to further classes of
connected graphs, namely, 2edge connected, simple and loopless graphs.
In the following, let graphs be loopless graphs with external edges allowed.
That is, edges which are connected to vertices only at one end. In the present
paper, we generalize formula (5) of [7] to biconnected graphs. Moreover, we give
a linear map to produce connected graphs from connected ones by increasing
the number of their biconnected components by one unit. We use this map
to give an algorithm to generate all equivalence classes of connected or 2edge
connected graphs with the exact coefficients. This is so that generated graphs
are automatically decomposed into their biconnected components. The proof
proceeds as suggested in [9]. That is, given an arbitrary equivalence class whose
representative is a graph on m internal edges, say, G, we show that every one of
the m internal edges of the graph G adds 1/(m·Aut(G)) to the sum of the co
efficients of all graphs isomorphic to G. To this end, we use the fact that vertices
carrying (labeled) external edges are held fixed under any automorphism.
This paper is organized as follows: Section 2 reviews the basic concepts of
graph theory underlying much of the paper. Section 3 contains the definitions of
the basic linear maps to be used in the following sections. Section 4.1 presents
an algorithm to generate biconnected graphs and gives some examples. Section
4.2 extends the result to connected and 2edge connected graphs.
2Basics
We briefly review the basic concepts of graph theory that are relevant for the
following sections. More details may be found in any standard textbook on
graph theory such as [2], or in [7] for the treatment of graphs with external
edges allowed. This section overlaps Section 2 of [7] except for the concept
of biconnected graph. However, as the present paper only considers loopless
graphs, the definition of graph given in that paper specializes here for loopless
graphs.
Let A and B denote sets. By [A,B], we denote the set of all unordered pairs
of elements of A and B, {{a,b}a ∈ A,b ∈ B}. In particular, by [A]2:= [A,A],
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we denote the set of all 2element subsets of A. Also, by 2A, we denote the
power set of A, i.e., the set of all subsets of A. By card(A), we denote the
cardinality of the set A. Furthermore, we recall that the symmetric difference
of the sets A and B is given by A△B := (A ∪ B)\(A ∩ B). Finally, given a
graph G, by Aut(G), we denote the order of its automorphism group Aut(G).
Let V = {vi}i∈Nand K = {ea}a∈Nbe infinite sets so that V ∩ K = ∅. Let
V ⊂ V; V ?= ∅ and K ⊂ K be finite sets. Let E = Eint∪ Eext ⊆ [K]2and
Eint∩Eext= ∅. Also, let the elements of E satisfy {ea,ea′}∩{eb,eb′} = ∅. That
is, ea,ea′ ?= eb,eb′. In this context, a graph is a triple G = (V,K,E) together
with the following maps:
(a) ϕint: Eint→ [V ]2;{ea,ea′} ?→ {vi,vi′};
(b) ϕext: Eext→ [V,K];{ea,ea′} ?→ {vi,ea′}.
The elements of V , E and K are called vertices, edges, and ends of edges,
respectively. In particular, the elements of Eint and Eext are called internal
edges and external edges, respectively. The degree of a vertex is the number of
ends of edges assigned to the vertex. Two distinct vertices connected together
by one or more internal edges, are said to be adjacent. Two or more internal
edges connecting the same pair of distinct vertices together, are called multiple
edges. Furthermore, let card(Eext) = s. Let L = {x1,...,xs} be a label set.
A labeling of the external edges of the graph G, is an injective map l : Eext→
[K,L];{ea,ea′} ?→ {ea,xz}, where z ∈ {1,...,s}. A graph G∗= (V∗,K∗,E∗);
E∗= E∗
graph G = (V,K,E); E = Eint∪ Eext, together with the maps ϕintand ϕextif
V∗⊆ V , K∗⊆ K, E∗⊆ E and ϕ∗
A path is a graph P = (V,K,Eint); V = {v1,...,vn}, n := card(V ) > 1, to
gether with the map ϕintso that ϕint(Eint) = {{v1,v2},{v2,v3},...,{vn−1,vn}}
and the vertices v1and vnhave degree 1, while the vertices v2,...,vn−1have
degree 2. In this context, the vertices v1and vnare called the end point ver
tices, while the vertices v2,...,vn−1 are called the inner vertices. A cycle is
a graph C = (V′,K′,E′
that ϕ′
has degree 2. A graph is said to be connected if every pair of vertices is joined
by a path. Otherwise, it is disconnected.
Given a graph G = (V,K,E); E = Eint∪ Eext, together with the maps ϕint
and ϕext, a maximal connected subgraph of the graph G is called a component.
Moreover, the set 2Eintis a vector space over the field Z2so that vector addition
is given by the symmetric difference. The cycle space C of the graph G is defined
as the subspace of 2Eintgenerated by all the cycles in G. The dimension of C is
called the cyclomatic number of the graph G. Let k := dimC, n := card(V ) and
m := card(Eint). Then, k = m−n+c, where c denotes the number of connected
components of the graph G [6].
Furthermore, given a connected graph, a vertex whose removal (together
with attached edges) disconnects the graph is called a cut vertex. A graph is
int∪ E∗
ext, together with the maps ϕ∗
intand ϕ∗
extis called a subgraph of a
int= ϕintE∗
int, ϕ∗
ext= ϕextE∗
ext.
int); V′= {v1,...,vn}, together with the map ϕ′
int) = {{v1,v2},{v2,v3},...,{vn−1,vn},{vn,v1}} and every vertex
intso
int(E′
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said to be 2connected (resp. 2edge connected) if it remains connected after
erasing any vertex (resp. any internal edge). A 2connected graph (resp. 2edge
connected graph) is also called biconnected (resp. edgebiconnected). Further
more, a biconnected component of a connected graph is a maximal biconnected
subgraph (see [1] 6.4 for instance).
Now, let L = {x1,...,xs} be a finite label set. Let G = (V,K,E); E =
Eint∪ Eext, card(Eext) = s, together with the maps ϕint and ϕext, and G∗=
(V∗,K∗,E∗); E∗= E∗
and ϕ∗
labelings of the elements of Eextand E∗
the graphs G and G∗is a bijection ψV : V → V∗and a bijection ψK: K → K∗
which satisfy the following three conditions:
int∪ E∗
ext, card(E∗
ext) = s, together with the maps ϕ∗
int
extdenote two graphs. Let l : Eext → [K,L] and l∗: E∗
ext→ [K∗,L] be
ext, respectively. An isomorphism between
(a) ϕint({ea,ea′}) = {vi,vi′} iff ϕ∗
int({ψK(ea),ψK(ea′)}) = {ψV(vi),ψV(vi′)};
(b) ϕext({ea,ea′}) = {vi,ea′} iff ϕ∗
ext({ψK(ea),ψK(ea′)}) = {ψV(vi),ψK(ea′)};
(c) L ∩ l({ea,ea′}) = L ∩ l∗({ψK(ea),ψK(ea′)}).
An isomorphism defines an equivalence relation on graphs. A vertex (resp.
edge) isomorphism between the graphs G and G∗is an isomorphism so that ψK
(resp. ψV) is the identity map. In this context, a symmetry of a graph G is an
isomorphism of the graph onto itself (i.e., an automorphism). A vertex symmetry
(resp. edge symmetry) of a graph G is a vertex (resp. edge) automorphism of the
graph. Given a graph G, let Autvertex(G) and Autedge(G) denote the groups of
vertex and edge automorphisms, respectively. Then, Aut(G) = Autvertex(G)·
Autedge(G) (a proof is given in [10] for instance).
3Elementary linear transformations
We introduce the elementary linear maps to be used in the following.
Given an arbitrary set X, by QX, we denote the free vector space on the set
X over Q. By idX: X → X;x ?→ x, we denote the identity map. Given maps
f : X → X∗and g : Y → Y∗, by [f,g], we denote the map [f,g] : [X,Y ] →
[X∗,Y∗];{x,y} ?→ {f(x),g(y)} with [f]2:= [f,f].
Let V = {vi}i∈N and K = {ea}a∈N be infinite sets so that V ∩ K = ∅.
Fix an integer s ≥ 0. Let L = {x1,...,xs} be a label set. For all integers
n ≥ 1 and k ≥ 0, by Wn,k,s
conn
(resp. Wn,k,s
connected graphs (resp. disconnected graphs with two components) with n
vertices, cyclomatic number k and s external edges whose free ends are labeled
x1,...,xs. In all that follows, let V = {v1,...,vn} ⊂ V, K = {e1,...,et} ⊂ K
and E = Eint∪ Eext ⊆ [K]2be the sets of vertices, ends of edges and edges,
respectively, of all elements of Wn,k,s
conn
(resp. Wn,k,s
n−1 (resp. card(Eint) = k+n−2) and card(Eext) = s. Also, let l : Eext→ [K,L]
be a labeling of their external edges. Finally, by Wn,k,s
disconn), we denote the set of all (loopless)
disconn), so that card(Eint) = k +
biconnand Wn,k,s
2edge, we denote
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the subsets of Wn,k,s
graphs, respectively.
conn
whose elements are biconnected and 2edge connected
We begin by briefly recalling the linear maps ξEext,V, li,jand si≥1introduced
in Sections 3 and 5.3.4 of [7]. We refer the reader to that paper for the precise
definitions.
(i) Distributing external edges between all elements of a given vertex subset in
all possible ways: Let G = (V,K,E); E = Eint∪ Eext, together with the
maps ϕintand ϕextdenote a graph in Wn,k,s
Let K′⊂ K be a finite set so that K ∩ K′= ∅. Also, let E′
s′:= card(E′
Assume that the elements of E′
{eb,eb′} = ∅. Let L′= {xs+1,...,xs+s′} be a label set so that L∩L′= ∅.
Let l′: E′
In′
E′
subsets: In′
E′
E′(j)
with
conn. Let V′= {vz1,...,vzn′} ⊆ V .
ext⊆ [K′]2;
ext).
extsatisfy {ea,ea′} ∩
ext→ [K′,L′] be a labeling of the elements of E′
extdenote the set of all ordered partitions of the set E′
ext= {(E′(1)
ext = ∅,∀i,j ∈ {1,...,n′}
ext. Finally, let
extinto n′disjoint
= E′
ext
and
ext ,...,E′(n′)
ext )E′(1)
ext ∪...∪E′(n′)
ext
E′(i)
ext∩
i ?= j}. In this context, the maps
ξE′
ext,V′ : QWn,k,s
conn
→ QWn,k,s+s′
conn
;G ?→
?
ext
(E′(1)
ext,...,E′(n′)
)∈In′
E′
ext
G(E′(1)
ext,...,E′(n′)
ext
)
are defined to produce each of the graphs G(E′(1)
G by assigning all elements of E′(j)
ext,...,E′(n′)
ext
)from the graph
ext to the vertex vzjfor all j ∈ {1,...,n′}.
(ii) The maps li,j: QWn,k,s
graph G∗from the graph G by connecting (or reconnecting) the vertices
viand vjwith an internal edge for all i,j ∈ {1,...,n} with i ?= j.
conn
→ QWn,k+1,s
conn
;G ?→ G∗are defined to produce the
(iii) (a) We define the maps si≥1from the maps sigiven in Section 3 of [7], by
restricting the image of the latter to graphs without isolated vertices.
More precisely, let G = (V,K,E); E = Eint∪ Eext, together with the
maps ϕintand ϕextdenote a graph in Wn,k,s
of ends of internal edges assigned to the vertex vi∈ V ; i ∈ {1,...,n}.
Let I2
nonempty disjoint sets: I2
E(2)
and
E(1)
denote the set of external edges assigned to the vertex vi. In this
context, for all i ∈ {1,...,n}, define the maps
conn. Let Eint,i⊂ K be the set
intdenote the set of all ordered partitions of the set Eint,iinto two
int= {(E(1)
int,i∩ E(2)
int,i,E(2)
int,i)E(1)
int,i,E(2)
int,i?= ∅,
E(1)
int,i∪
int,i= Eint,i
int,i= ∅}. Moreover, let Lext,i ⊂ Eext
si≥1: QWn,k,s
conn
→ QWn+1,k−1,s
conn
∪ QWn+1,k,s
disconn;
0 ≤ card(Eint,i) < 2;
G ?→
0if
ξLext,i,{vi,vn+1}
??
(E(1)
int,i,E(2)
int,i)∈I2
intG(E(1)
int,i,E(2)
int,i)
?
otherwise,
where each of the graphs G(E(1)
as follows: (a) split the vertex vi into two vertices, namely, vi and
int,i,E(2)
int,i)is produced from the graph G
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