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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, F-75015 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 2-edge connected graphs. The underlying algorithms are

amenable to computer implementation.

1 Introduction

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 19th-century [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, 1649-003 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, 2-edge 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 2-edge

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 2-edge connected graphs.

2 Basics

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 2-element 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= ϕint|E∗

int, ϕ∗

ext= ϕext|E∗

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 2-connected (resp. 2-edge connected) if it remains connected after

erasing any vertex (resp. any internal edge). A 2-connected graph (resp. 2-edge

connected graph) is also called biconnected (resp. edge-biconnected). 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

2-edge, we denote

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the subsets of Wn,k,s

graphs, respectively.

conn

whose elements are biconnected and 2-edge 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

non-empty 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 ?→

0 if

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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vn+1; (b) assign the ends of edges in E(1)

respectively.

(b) Let Bi= {G1

nents of the graph G containing the vertex vi. That is, vi∈ Vj

Vj

Kjdenote the set of ends of edges of the graph Gj

be the set of ends of edges assigned to the vertex viwhich belong to

the graph Gj

j,m = 1,...,niwith j ?= m. In this context, we define the maps ˆ si≥1

by restricting the image of si≥1to graphs obtained by replacing in

the above definition the set I2

ˆE(1)

∅,ˆE(2)

so that card(Ej

int,iand E(2)

int,ito viand vn+1,

i,...,Gni

i} with ni≥ 1, be the set of biconnected compo-

i, where

idenotes the vertex set of the graph Gj

ifor all j ∈ {1,...,ni}. Let

i. Let Ej

int,i⊂ Kj

i. Hence, Eint,i= ∪ni

j=1Ej

int,iand Ej

int,i∩ Em

int,i= ∅ for all

intbyˆI2

int,i∩ˆE(2)

int= {(ˆE(1)

int,i,ˆE(2)

and

int,i)|ˆE(1)

ˆE(1)

int,i,ˆE(2)

int,i∩ Ej

int,i?=

∅,

int,i∪ˆE(2)

int,i∩ Ej

int,i= Eint,i,ˆE(1)

int,i= ∅

int,i?=

int,i?= ∅∀j = 1,...,ni} with ˆ si≥1(G) := 0 if there exists j

int,i) < 2.

(c) We define the maps qi

1

2lρ

(ρ)

≥1and ˆ qi

(ρ)

≥1in analogy with the maps q(ρ)

i

:=

i,n+1◦ sigiven in [7] (see also [3] for ρ = 1):

qi

(ρ)

≥1:=

1

2(ρ − 1)!lρ

1

2(ρ − 1)!lρ

i,n+1◦ si≥1: QWn,k,s

conn

→ QWn+1,k+ρ−1,s

conn

, (1)

ˆ qi

(ρ)

≥1:=

i,n+1◦ ˆ si≥1: QWn,k,s

conn

→ QWn+1,k+ρ−1,s

conn

, (2)

where lρ

i,n+1denotes the ρth iterate of li,n+1with l0

We now introduce the following auxiliary map:

i,n+1= id.

(iv) Fix integers n′> 0 and 1 ≤ i ≤ n′. Let G = (V,K,Eint) together with

the map ϕintdenote a graph in Wn,k,0

where the graph G∗satisfies the following conditions:

conn. We define the map Ξi: G ?→ G∗,

(a) V∗= χi

v(V ), where χi

v: vl ?→

?

vi

if

l ∈ {2,...,n}

l = 1

vn′+l−1

if

is a

bijection;

(b) K∗= χt

(c) E∗

(d) ϕ∗

e(K), where χt

e]2(Eint);

e]2= [χi

e: eb?→ et+bis a bijection;

int= [χt

int◦ [χt

v]2◦ ϕint.

The maps Ξiare extended to the whole of QWn,k,0

conn

by linearity.

(v) (a) Distributing the biconnected components of a connected graph sharing

a vertex, between all the vertices of a given biconnected graph in all

possible ways: Let G = (V,K,E); E = Eint∪ Eext, together with

the maps ϕint and ϕext denote a graph in Wn,k,s

and Lext,i ⊆ Eext be the sets of internal and external edges, re-

spectively, assigned to the vertex vi ∈ V with i ∈ {1,...,n}. Let

conn. Let Lint,i ⊆ Eint

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Bi = {G1

graph G sharing the vertex vi. That is, vi∈ Vj

the vertex set of the graph Gj

the set of all ordered partitions of the set Bi into n′disjoint sets:

In′

i

)|B(1)

i

∅,∀l,l′∈ {1,...,n′}

withl ?= l′}.

B(l)

i

= {G(l)

of the graph G(l)

V′(l)

a

⊆ V(l)

?n(l)

Wn′,k′,0

be the graph obtained from G′by applying the map Ξigiven above.

In this context, for all i ∈ {1,...,n} define

i,...,Gni

i} be the set of biconnected components of the

i, where Vj

idenotes

Bidenote

ifor all j ∈ {1,...,ni}. Let In′

Bi= {(B(1)

i,...,B(n′)

∪ ... ∪ B(n′)

i

= Bi

For all l ∈ {1,...,n′}, let

and

B(l)

i

∩ B(l′)

i

=

i,1,...,G(l)

i,afor all a ∈ {1,...,n(l)}. Let L(l)

i,a\{vi} be so that ϕint(L(l)

int,i,aand V′(l):=?n(l)

i,n(l)}. Also, let V(l)

i,a⊂ V denote the vertex set

int,i,a⊆ Lint,i and

]. Let L(l)

int,i,a) = [vi,V′(l)

. Finally, let G′denote a graph in

a

int,i:=

a=1L(l)

biconn . Also, let Ξi(G′) = (ˆV ,ˆK,ˆEint) together with the map ˆ ϕint

a=1V′(l)

a

rG′

i

: QWn,k,s

conn → QWn+n′−1,k+k′,s

conn

?

(B(1)

i

;

G ?→ ξLext,i,ˆV

?

,...,B(n′)

i

)∈In′

Bi

G(B(1)

i

,...,B(n′)

i

)

?

,

where the graphs G(B(1)

together with the maps ϕ∗

(a) V∗= V \{vi} ∪ˆV ;

(b) K∗= K ∪ˆK;

(c) E∗= E∗

(d) ϕ∗

ϕ∗

l ∈ {2,...,n′};

(e) ϕ∗

(f) l∗= l|Eext\Lext,iis a labeling of the elements of E∗

The maps rG′

i

are extended to the whole of QWn,k,s

For instance, let C4denote a cycle on four vertices. Figure 1 shows

the result of applying the map rC4

connected graph with two biconnected components.

i

,...,B(n′)

intand ϕ∗

i

)= (V∗,K∗,E∗); E∗= E∗

extsatisfy the following conditions:

int∪ E∗

ext,

int∪ E∗

ext, where E∗

int= Eint∪ˆEint, E∗

int|ˆ Eint= ˆ ϕint|ˆ Eint;

int(L(l)

ext= Eext\Lext,i;

int|Eint\Lint,i= ϕint|Eint\Lint,i, ϕ∗

int(L(1)

int,i) = [vi,V′(1)] and ϕ∗

int,i) = [vl+n′−1,V′(l)] for all

ext|Eext\Lext,i= ϕext|Eext\Lext,i;

ext.

conn

by linearity.

cutvertexto the cut vertex of a 2-edge

= 2

)

(

2

+

R3,4◦ △3

Figure 1: Linear combination of graphs obtained by applying the maps rC4

to the cut vertex of the graph consisting of two triangles sharing one vertex.

cutvertex

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(b) We define the maps ˆ riGby restricting the image of rG

tained by replacing in the definition of the latter the set In′

lowing set of n′-tuples: {(Bi,∅,...,∅),(∅,Bi,...,∅),...,(∅,...,∅,Bi)}.

(c) Given a linear combination of graphs ϑ =?

in QWn,k,s

rϑ

i:=

?

G∈Wn,k,s

ito graphs ob-

Biby the fol-

G∈Wn,k,s

biconnαGG; αG∈ Q

biconn, define

biconn

αGrG

i. (3)

ˆ riϑ:=

?

biconn

G∈Wn,k,s

αGˆ riG. (4)

We proceed to generalize the edge contraction operation to the operation

of contracting a biconnected component of a connected graph.

(vi) Contracting a biconnected component of a connected graph: Let G =

(V,K,E); E = Eint∪ Eext, together with the maps ϕint and ϕext denote

a graph in Wn,k,s

component of the graph G:

ˆG = (ˆV ,ˆK,ˆE);ˆV = {vi1,...,vin′} ⊆ V ,

ˆK = {ej1,...,ejt′} ⊆ K;ˆE =ˆEint∪ˆEext⊆ E, together with the maps ˆ ϕint

and ˆ ϕext. Let i1< ... < in′ and j1 < ... < jt′. Also, let k′denote the

cyclomatic number of the graphˆG. Let B = {Gl,...,GnB} be the set of

biconnected components of the graph G so thatˆV ∩Vl?= ∅, where Vl⊂ V

denotes the vertex set of the graph Gl for all l ∈ {1,...,nB}. Now, let

Lint,l⊆ Eint and V′

l ∈ {1,...,nB}. Also, let LB

let Efree⊂ K denote the set of free ends of the external edges inˆEext. In

this context, define

conn, where n > 1. Consider the following biconnected

l⊆ Vl\(ˆV ∩ Vl) be so that ϕint(Lint,l) := [ˆV ∩ Vl,V′

int:=?nB

l];

l=1Lint,land V′B:=?nB

l=1V′

l. Finally,

cˆ G: QWn,k,s

conn

→ QWn−n′+1,k−k′,s

conn

;G ?→ G∗,

where the graph G∗= (V∗,K∗,E∗); E∗= E∗

maps ϕ∗

int∪ E∗

ext, together with the

intand ϕ∗

extsatisfies the following conditions:

(a) V∗= χn′

v(V \(ˆV \{vi1})), where

χn′

v: vl?→

vl

ifl ∈ {1,...,i2− 1}

vl−j+1

vl−n′+1

ifl ∈ {ij+ 1,...,ij+1− 1},j ∈ {2,...,n′}

ifl ∈ {in′ + 1,...,n}

is a bijection;

(b) K∗= χt′

e(K\ˆK), where

χt′

e: eb?→

eb

ifb ∈ {1,...,j1− 1}

eb−l

eb−t′

ifb ∈ {jl+ 1,...,jl+1− 1},l ∈ {2,...,t′− 1}

ifb ∈ {jt′ + 1,...,t}

is a bijection;

8

Page 9

(c) E∗= E∗

(d) ϕ∗

ϕ∗

(e) ϕ∗

ϕ∗

int∪ E∗

e]2|Eint\(ˆ Eint∪LB

int([χt′

ext◦ [χt′

ext([χt′

ext, where E∗

int= [χt′

int)= [χn′

v(V′B)];

v,χt′

e]2(Eint\ˆEint), E∗

v]2◦ ϕint|Eint\(ˆ Eint∪LB

ext= [χt′

e]2(Eext);

int◦ [χt′

int);

e]2(LB

e]2|Eext\ˆ Eext= [χn′

e]2(ˆEext)) = [vi1,χt′

int) = [vi1,χn′

e] ◦ ϕext|Eext\ˆ Eext;

e(Efree)];

−1]2: E∗

(f) l∗= [χt′

of E∗

e,idL]◦l◦[χt′

ext.

e

ext→ [K∗,L] is a labeling of the elements

(vii) Erasing all external edges of a connected graph: Let G = (V,K,E); E =

Eint∪Eext, together with the maps ϕintand ϕextdenote a graph in Wn,k,s

Let K1,K2⊂ K satisfy K1∩ K2= ∅ and Eext= [K1,K2]. Let K1∪ K2=

{ei1,...,ei2s} with i1< ... < i2s. In this context, define

conn.

ψ : QWn,k,s

conn

→ QWn,k,0

conn;G ?→ G∗,

where the graph G∗= (V∗,K∗,E∗

the following conditions:

int) together with the map ϕ∗

intsatisfies

(a) V∗= V ;

(b) K∗= χ2s

(c) E∗= [χ2s

(d) ϕ∗

e(K\(K1∪ K2)), where the bijection χ2s

e]2(Eint);

e]2(Eint)) = ϕint.

e is given in (vi);

int([χ2s

The following lemmas are now established.

Lemma 1. Fix integers s ≥ 0, k > 0 and n > 1. Then, for all i ∈ {1,...,n},

qi

biconn

.

(ρ)

≥1(QWn,k,s

biconn) ⊆ QWn+1,k+ρ−1,s

Proof. Let G denote a graph in Wn,k,s

si≥1to the graph G. si≥1(G) is a linear combination of connected graphs.

Therefore, by Lemma 5 of [7] the graphs in qi

Clearly, they must be biconnected in particular.

biconn. Apply the map qi

(ρ)

≥1:=

1

2(ρ−1)!lρ

i,n+1◦

(ρ)

≥1(G) are 2-edge connected.

Lemma 2. Fix integers s ≥ 0, k,> 0 and n > 1. (a) Let G denote a graph

in Wn,k,s

Let G denote a graph in Wn,k,s

(ρ)

≥1(G) ∈ Wn+1,k+ρ−1,s

2-edge

\Wn+1,k+ρ−1,s

biconn

.

2-edge with only one cut vertex. Then, ˆ q(ρ)

cutvertex≥1(G) ∈ Wn+1,k+ρ−1,s

biconn

. (b)

2-edge with at least two cut vertices. Then, qi

(ρ)

≥1(G)−

ˆ qi

Proof. The proof is trivial.

Lemma 3. Fix integers s ≥ 0, k,k′> 0 and n,n′> 1. Let G denote a graph

in Wn′,k′,0

biconn . Then, for all i ∈ {1,...,n}, rG

i(QWn,k,s

2-edge) ⊆ QWn+n′−1,k+k′,s

2-edge

.

Proof. The proof is trivial.

9

Page 10

4 Recursion formulas

We give a recursion formula to generate all equivalence classes of biconnected

graphs. In a recursion step, the formula yields a linear combination of bicon-

nected graphs with the same vertex and cyclomatic numbers. The key feature is

that the sum of the (rational) coefficients of all graphs in the same equivalence

class corresponds to the inverse of the order of their group of automorphisms.

We extend the result to connected or 2-edge connected graphs. The underlying

algorithms are amenable to direct implementation via coalgebra in the sense of

[8].

4.1Biconnected graphs

We pick out the terms that generate biconnected graphs in formula (5) of [7].

Theorem 4. Fix an integer s ≥ 0. For all integers k ≥ 0 and n > 1, define

βn,k,s

biconn ∈ QWn,k,s

biconn by the following recursion relation:

• β2,k,s

on two vertices;

biconn :=

1

2(k+1)!ξEext,{v1,v2}(lk

1,2(P2)), where P2 ∈ W2,0,0

biconn denotes a path

• βn,0,s

biconn:= 0, n > 2;

•

βn,k,s

biconn:=

1

k + n − 1

?k+1

ρ=1

?

n−1

?

i=1

qi

(ρ)

≥1(βn−1,k+1−ρ,s

biconn

)+

n−2

?

j=2

k−j+1

?

ρ=1

ˆ q(ρ)

cutvertex≥1(βn−1,k+1−ρ,s

j

)

?

, (5)

where for all integers j > 1, n ≥ j+1 and k ≥ j, βn,k,s

recursion relation:

j

is given by the following

•

βn,k,s

2

:=

1

k + n − 1

k−1

?

k′=1

n−1

?

n′=2

n−n′+1

?

i=1

?

(k′+n′−1)rβn′,k′,0

biconn

i

(βn−n′+1,k−k′,s

biconn

)

?

(6)

;

•

βn,k,s

j

:=

1

k + n − 1

k−1

?

k′=1

n−1

?

n′=2

?

(k′+n′−1)ˆ rβn′,k′,0

biconn

cutvertex (βn−n′+1,k−k′,s

j−1

)

?

. (7)

10

Page 11

Then, for fixed values of n and k, βn,k,s

G ∈ Wn,k,s

following holds: (i) There exists G ∈ C so that αG > 0; (ii)

1/|Aut(C)|.

biconn =?

G∈Wn,k,s

biconnαGG; αG ∈ Q for all

biconn. Moreover, given an arbitrary equivalence class C ⊆ Wn,k,s

biconn, the

G∈CαG =

?

In formula (5), the ˆ q(ρ)

k < 2.

cutvertex≥1summand does not appear when n < 4 or

Proof. Note that βn,k,s

2-edge connected graphs with only one cut vertex, j biconnected components,

n vertices, cyclomatic number k and s external edges, with coefficient given by

the inverse of the order of their automorphism group. This is a particular case

of Theorem 5 to be given later on. We refer the reader to the next section

for the proof. Thus, by Lemmas 1 and 2, the proof of Theorem 4 follows

straightforwardly from that of Theorem 18 of [7].

j

is the linear combination of all equivalence classes of

4.1.1Examples

We show the result of computing all mutually non-isomorphicbiconnected graphs

without external edges as contributions to βn,k,0

2 ≤ n + k ≤ 6. The coefficients in front of graphs are the inverses of the orders

of their groups of automorphisms.

biconnvia formula (5) up to order

n = 2,k = 0

1

2

n = 2,k = 1

1

22

n = 3,k = 1

1

3!

n = 2,k = 2

1

2·3!

n = 4,k = 1

1

8

11

Page 12

1

22

n = 3,k = 2

n = 2,k = 3

1

2·4!

1

10

n = 5,k = 1

n = 4,k = 2

1

22

+

1

22

1

23

n = 3,k = 3

+

1

2·3!

n = 2,k = 4

1

2·5!

4.2Connected graphs

We use the maps rβn,k,s

given by formulas (3) and (5), respectively, to generate all equivalence classes

of connected graphs. The underlying algorithm is so that generated graphs are

automatically decomposed into their biconnected components.

biconn

i

and the linear combination of graphs βn,k,s

biconn∈ QWn,k,s

biconn

Theorem 5. Fix an integer s ≥ 0. For all integers k ≥ 0 and n > 1, define

βn,k,s

conn

∈ QWn,k,s

conn

by the following recursion relation:

:= β2,k,s

• β2,k,s

conn

biconn;

•

βn,k,s

conn

:= βn,k,s

biconn+

1

k + n − 1·

12

Page 13

k

?

k′=0

n−1

?

n′=2

n−n′+1

?

i=1

?

(k′+ n′− 1)rβn′,k′,0

biconn

i

(βn−n′+1,k−k′,s

conn

)

?

,n > 2.(8)

Then, for fixed values of n and k, βn,k,s

G ∈ Wn,k,s

following holds: (i) There exists G ∈ C so that αG > 0; (ii)

1/|Aut(C)|.

conn

=?

G∈Wn,k,s

conn

αGG; αG ∈ Q for all

conn . Moreover, given an arbitrary equivalence class C ⊆ Wn,k,s

conn, the

G∈CαG =

?

Proof. Let G denote any connected graph with m ≥ 1 internal edges.

proceed to show that an arbitrary biconnected component of the graph G with

m′≥ 1 internal edges adds m′/(m · |Aut(G)|) to the sum of the coefficients of

all graphs isomorphic to the graph G. As expected, we thus conclude that every

one of the m internal edges of the graph G contributes 1/(m·|Aut(G)|) to that

sum.

We

Lemma 6. Fix integers s ≥ 0, k ≥ 0 and n > 1. Let βn,k,s

QWn,k,s

conn

be defined by formula (8). Let C ⊆ Wn,k,s

alence class. Then, there exists G ∈ C so that αG> 0.

conn

denote an arbitrary equiv-

=?

G∈Wn,k,s

conn

αGG ∈

conn

Proof. The proof proceeds by induction on the number of biconnected compo-

nents µ. By Theorem 4, the statement holds for all graphs in βn,k,s

one biconnected component. We assume the statement to hold for graphs in

βn,k,s

conn

with µ − 1 ≥ 1 biconnected components. Let G denote any graph in

C ⊆ Wn,k,s

conn

ηG′ ∈ Q be given by equation (5). Recall that by equation (3) the maps rβn′,k′,0

read as

rβn′,k′,0

i

:=

?

G′∈Wn′,k′,0

conn with only

with µ biconnected components. Let βn′,k′,0

biconn =?

G′∈Wn′,k′,0

biconnηG′G′;

biconn

i

biconn

biconn

ηG′ rG′

i . (9)

We proceed to show that a graph isomorphic to G is generated by applying

the maps rβn′,k′,0

i

to graphs with µ − 1 biconnected components occurring in

βn−n′+1,k−k′,s

conn

=

?

cient.LetˆG be an arbitrary biconnected component of the graph G.

ˆV = {vi1,...,vin′} ⊂ V be its vertex set, where i1 < ... < in′. Also, let k′

denote its cyclomatic number. Contracting the graphˆG to the vertex vi1yields

a graph cˆ G(G) ∈ Wn−n′+1,k−k′,s

conn

with µ − 1 biconnected components. Let B

denote the equivalence class containing the graph cˆ G(G). Let G′∈ Wn′,k′,0

be a biconnected graph isomorphic to ψ(ˆG), where ψ(ˆG) is the graph obtained

fromˆG by erasing all the external edges. By induction assumption, there ex-

ists a graph in B, say, H∗, so that H∗ ∼= cˆ G(G) and βH∗ > 0. Let vj with

j ∈ {1,...,n − n′+ 1} be the vertex of the graph H∗which is mapped to vi1

of cˆ G(G) by an isomorphism. Applying the map rG′

linear combination of graphs, one of which, say, H, is isomorphic to G. That is,

αH> 0 and H∼= G.

biconn

G∗∈Wn−n′+1,k−k′,s

conn

βG∗G∗; βG∗ ∈ Q with non-zero coeffi-

Let

biconn

j

to the graph H∗yields a

13

Page 14

Lemma 7. Fix integers k ≥ 0, n > 1 and s ≥ n. Let C ⊆ Wn,k,s

equivalence class. Let G = (V,K,E); E = Eint∪ Eext, together with the maps

ϕintand ϕextdenote a graph in C. Assume that V ∩ϕext(Eext) = V . Let βn,k,s

?

1/|Aut(C)|.

conn

denote an

conn

=

G∈Wn,k,s

connαGG ∈ QWn,k,s

conn

be defined by formula (8). Then,

?

G∈CαG =

Proof. The proof proceeds by induction on the number of biconnected compo-

nents µ. By Theorem 4, the statement holds for all graphs in βn,k,s

one biconnected component. We assume the statement to hold for graphs in

βn,k,s

have µ biconnected components. Let m = k + n − 1 denote its internal edge

number. By Lemma 6, there exists a graph, say, H ∈ C which occurs in βn,k,s

with non-zero coefficient. That is, H∼= G and αH > 0. Moreover, the graph

H ∈ C is so that every one of its vertices has at least one (labeled) external

edge. Hence, |Autvertex(C)| = 1 so that |Aut(C)| = |Autedge(C)|. We proceed

to show that?

with µ − 1 biconnected components, the graphs in the equivalence class C are

generated by the recursion formula (8), and how many times they are generated.

Choose any one of the µ biconnected components of the graph H ∈ C. Let

this be a graph, say,ˆG, with vertex setˆV = {vi1,...,vin′} ⊆ V , where i1 <

... < in′. Also, let k′, m′= k′+n′−1 and s′≥ n′denote its cyclomatic number,

internal edge number and external edge number, respectively. Moreover, let A

denote the equivalence class containing the graphˆG. Since this is a subgraph of

the graph H, |Aut(A )| = |Autedge(A )|. Contracting the graphˆG to the vertex

vi1yields a graph cˆ G(H) ∈ Wn−n′+1,k−k′,s

conn

Let B denote the equivalence class containing cˆ G(H). The graphs in B have no

non-trivial vertex symmetries. Hence, the order of their automorphism group is

related to that of the graph H ∈ C via

conn

with only

conn with µ − 1 ≥ 1 biconnected components. Let the graph G ∈ C ⊆ Wn,k,s

conn

conn

G∈CαG= 1/|Aut(C)|. To this end, we check from which graphs

with µ−1 biconnected components.

|Aut(B)| =|Aut(C)|

|Aut(A )|.

Let βn−n′+1,k−k′,s

conn

sumption,

=?

G∗∈Wn−n′+1,k−k′,s

conn

βG∗G∗;βG∗ ∈ Q. By induction as-

?

G∗∈B

βG∗ =

1

|Aut(B)|.

Now, let βn′,k′,0

connected graph isomorphic to ψ(ˆG). Let D ⊆ Wn′,k′,0

class containing G′. The order of the automorphism group of the graph G′is

related to that ofˆG via

biconn

=?

G′∈Wn′,k′,0

biconnηG′G′; ηG′ ∈ Q. Let G′∈ Wn′,k′,0

biconn

be a bi-

biconn

denote the equivalence

|Aut(D)| = |Aut(A )| · |Autvertex(D)|

for |Aut(A )| = |Autedge(A )| = |Autedge(D)|. By Lemma 6, there exists a

graph, say, H∗∈ B so that H∗ ∼= cˆ G(H) and βH∗ > 0. Let vj with j ∈

14

Page 15

{1,...,n − n′+ 1} be the vertex of the graph H∗which is mapped to vi1of

cˆ G(H) by an isomorphism. Apply the map rG′

there are |Autvertex(D)| ways to distribute the s′external edges assigned to

the vertex vj of the graph H∗between all the vertices of the graph Ξj(G′) so

as to obtain a graph in the equivalence class C ∋ H. Therefore, there are

|Autvertex(D)| graphs in the linear combination rG′

to the graph G. Clearly, the map rG′

j

produces a graph isomorphic to H from

the graph H∗with coefficient α∗

H= βH∗ ∈ Q. Now, formula (8) prescribes to

apply the maps rG′

i

to the vertex which is mapped to vi1by an isomorphism of

every graph in the equivalence class B occurring in βn−n′+1,k−k′,s

coefficient. Therefore,

j

to the graph H∗. Notice that

j(H∗) which are isomorphic

conn

with non-zero

?

G∈C

α∗

G

=

|Autvertex(D)| ·

?

G∗∈B

βG∗

=

|Autvertex(D)|

|Aut(B)|

|Autvertex(D)| · |Aut(A )|

|Aut(C)|

|Aut(D)|

|Aut(C)|,

=

=

where the factor |Autvertex(D)| on the right hand side of the first equality, is

due to the fact that every graph (with non-zero coefficient) in the equivalence

class B generates |Autvertex(D)| graphs in C. Hence, according to Theorem 4

and formulas (8) and (9), the contribution to?

Distributing this factor between the m′internal edges of the graphˆG yields

1/(m · |Aut(C)|) for each edge. Repeating the same consideration for every bi-

connected component of the graph G yields that every edge of each biconnected

component adds 1/(m · |Aut(C)|) to?

We conclude that every one of the m internal edges of the graph G con-

tributes 1/(m·|Aut(C)|) to?

1/|Aut(C)|. This completes the proof.

G∈CαGis m′/(m · |Aut(C)|).

G∈CαG.

G∈CαG. Hence, the overall contribution is exactly

βn,k,s

conn satisfies the following property.

Lemma 8. Fix integers s ≥ 0, k ≥ 0 and n > 1. Let βn,k,s

QWn,k,s

conn

be defined by formula (8). Moreover, let K′⊂ K be a finite set so that

K ∩ K′= ∅. Let E′

E′

set so that L ∩ L′= ∅. Let l′: E′

E′

conn

= ξE′

conn

=?

G∈Wn,k,s

conn

αGG ∈

ext⊆ [K′]2; s′:= card(E′

ext). Assume that the elements of

extsatisfy {ea,ea′} ∩ {eb,eb′} = ∅. Also, let L′= {xs+1,...,xs+s′} be a label

ext→ [K′,L′] be a labeling of the elements of

ext. Then, βn,k,s+s′

ext,V(βn,k,s

conn).

Proof. Let V∗= {vi,vn−n′+2,...,vn} ⊆ V be the vertex set of all graphs in

Ξi(βn′,k′,0

. Furthermore, ξLext,V∗ ◦ rβn′,k′,0

biconn ); i ∈ {1,...,n−n′+1}. Clearly, ξE′

ext,V=?

Lext⊆E′

extξE′

ext\Lext,V \V∗◦

= rβn′,k′,0

i

ξLext,V∗ : QWn,k,s

conn

→ QWn,k,s+s′

conn

biconn

i

biconn

◦

15

Page 16

ξLext,{vi}: QWn−n′+1,k−k′,s

the equality βn,k,s+s′

definition (8).

conn

→ QWn,k,s+s∗

conn

ext,V(βn,k,s

, where s∗= card(Lext). Therefore,

conn

= ξE′

conn) follows immediately from the recursive

Lemma 9. Fix integers s ≥ 0, k ≥ 0 and n > 1. Let C ⊆ Wn,k,s

arbitrary equivalence class. Let βn,k,s

conn

by formula (8). Then,?

Proof. The proof is the same as that of Lemma 10 of [7] (see also Lemma 10

and Theorem 10 of [9] and [10], respectively).

conn

denote an

be defined=?

G∈Wn,k,s

conn

αGG ∈ QWn,k,s

conn

G∈CαG= 1/|Aut(C)|.

This completes the proof of Theorem 5.

4.2.1Examples

The present section overlaps Section 5.3.3 of [7]. We show the result of com-

puting all mutually non-isomorphic connected graphs without external edges

as contributions to βn,k,0

conn

via formula (8) up to order 2 ≤ n + k ≤ 5. The

coefficients in front of graphs are the inverses of the orders of their groups of

automorphisms.

n = 2,k = 0

1

2

n = 3,k = 0

1

2

n = 2,k = 1

1

22

n = 4,k = 0

+

1

3!

1

2

n = 3,k = 1

1

3!

+1

2

n = 2,k = 2

1

2·3!

16

Page 17

n = 5,k = 0

1

2

+

1

4!

+1

2

+

n = 4,k = 1

+1

4

+

+1

2

+1

4

1

8

+1

2

n = 3,k = 2

+

1

23

1

22

+

1

3!

n = 2,k = 3

1

2·4!

4.3 2-edge connected graphs

By Lemma 3, Theorem 5 generalizes straightforwardly to 2-edge connected

graphs.

Theorem 10. Fix an integer s ≥ 0. For all integers k > 0 and n > 1, define

βn,k,s

2-edge∈ QWn,k,s

2-edge by the following recursion relation:

• β2,k,s

2-edge:= β2,k,s

biconn;

•

βn,k,s

2-edge:= βn,k,s

biconn+

1

k + n − 1·

n−n′+1

?

i=1

k−1

?

k′=1

n−1

?

n′=2

?

(k′+ n′− 1)rβn′,k′,0

biconn

i

(βn−n′+1,k−k′,s

2-edge

)

?

,n > 2. (10)

Then, for fixed values of n and k, βn,k,s

2-edge =?

G∈Wn,k,s

2-edgeαGG; αG ∈ Q for all

G ∈ Wn,k,s

2-edge. Moreover, given an arbitrary equivalence class C ⊆ Wn,k,s

2-edge, the

17

Page 18

following holds: (i) There exists G ∈ C so that αG > 0; (ii)?

1/|Aut(C)|.

G∈CαG =

4.3.1Examples

We show the result of computing all mutually non-isomorphic 2-edge connected

graphs without external edges as contributions to βn,k,0

order 3 ≤ n + k ≤ 6. The coefficients in front of graphs are the inverses of the

orders of their groups of automorphisms.

2-edgevia formula (10) up to

n = 2,k = 1

1

22

n = 3,k = 1

1

3!

n = 2,k = 2

1

2·3!

n = 4,k = 1

1

8

+

1

23

1

22

n = 3,k = 2

n = 2,k = 3

1

2·4!

1

10

n = 5,k = 1

18

Page 19

n = 4,k = 2

+

1

22

1

22

+

1

22

1

23

n = 3,k = 3

+

1

2·3!

+

1

2·3!

n = 2,k = 4

1

2·5!

4.3.2Algorithmic considerations

We briefly discuss some of the algorithmic implications of the recursive definition

(10). In the present section, only graphs without external edges are considered

for these may be added via the maps ξEext,V.

An important algorithmic aspect is to determine a priori the nature of the

biconnected components of the 2-edge connected graphs generated by formula

(10). In this context, the most straightforward simplification is to restrict the

formula to graphs whose biconnected components are cycles. Let Cndenote a

cycle with n vertices. Clearly, from formula (5) βn,1,0

simplicity all graphs in the same equivalence class are identified as the same.

Therefore, formula (10) specializes to 2-edge connected graphs with the aforesaid

property as follows:

biconn= 1/(2n)Cn, where for

β2,1,0

2-edge,C

:=

1

22C2;

βn,k,0

2-edge,C

:=

1

2nCn+

1

2(k + n − 1)

n−1

?

n′=2

n−n′+1

?

i=1

rCn′

i

(βn−n′+1,k−1,0

2-edge,C

),n > 2,

where βn,k,0

graphs with n vertices and cyclomatic number k so that every biconnected com-

ponent is a cycle. In addition, suppose that one is only interested in calculating

all 2-edge connected graphs whose biconnected components have a minimum

vertex or cyclomatic number, say, nmin and kmin, respectively. This is clearly

obtained by changing the lower and upper limits of the sums over n′and k′in

formula (10) to nminand n − nmin+ 1 or kminand k-kmin, respectively.

2-edge,C∈ QWn,k

2-edgedenotes the linear combination of all 2-edge connected

Acknowledgements

The author would like to thank Brigitte Hiller and Christian Brouder for careful

reading of the manuscript. The research was supported through the fellowship

19

Page 20

SFRH/BPD/48223/2008 provided by the Portuguese Science and Technology

Foundation.

References

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