A Short Proof of Twin-Quotient Decomposition and Orbit Lifting

1. Introduction

Twin classes are elementary examples of modules in the modular decomposition of graphs. See Gallai [1] and the survey of Habib and Paul [2]. Automorphism groups of graph compositions and lexicographic products were studied by Sabidussi [3,4].

The group decomposition proved below is known. Tadayyonfar and Ashrafi ([5] [Theorem 1.2]) give the corresponding semidirect product for reduced complete-empty X-joins, with connected X in their setup. Here G is obtained by substituting the clique or independent graph $G\left[C\right]$ at each vertex C of its twin quotient. This substitution is reduced: if two quotient vertices are nonadjacent twins and both substituted graphs are edgeless, their union would be a twin class of G; the same contradiction arises for adjacent quotient twins whose substituted graphs are both complete. Their type-preserving quotient group is precisely the group preserving the colours used here. Thus, for connected G, Corollary 1 is a direct instance of their theorem. Our proof uses the invariance of maximal twin classes and applies without a connectedness assumption.

Cameron [6] studies iteration to a twin-free graph and obtains a normal series with factors that are products of symmetric groups, together with an embedding of the remaining quotient group into the automorphism group of the reduced graph. Here we retain the coloured quotient after one contraction step, which Remark 1 demonstrates is not in general twin-free. The purpose of this note is a self-contained short proof of the decomposition and an explicit account of its vertex- and edge-orbit consequences, including the bounds supplied by partial quotient symmetry information.

2. Definitions

Throughout, $G=(V,E)$ is a finite simple undirected graph, with $E\subseteq \left({\displaystyle \genfrac{}{}{0pt}{}{V}{2}}\right)$. Here $\left({\displaystyle \genfrac{}{}{0pt}{}{S}{2}}\right)$ denotes the set of unordered two-element subsets of a set S. For $S\subseteq V$, let $G\left[S\right]$ denote the subgraph of G induced on S, and for disjoint $S,T\subseteq V$ let $G[S,T]$ denote the bipartite subgraph with parts S and T whose edges are the edges of G having one endpoint in S and the other in T. For $v\in V$, let $N_G\left(v\right)$ denote its open neighbourhood. Define

$$u{\equiv }_{T}v⟺u=v\mathrm{or}{N}_G\left(u\right)\setminus \left\{v\right\}=N_G\left(v\right)\setminus \left\{u\right\}.$$

(1)

The equivalence classes of this relation, established below, are called the maximal twin classes and form a partition $\mathcal{C}$ of V.

For a group L acting on a set X, write

$$Lx=\{\ell \left(x\right):\ell \in L\},{\mathcal{O}}_L\left(X\right)=\{Lx:x\in X\}.$$

An action on vertices induces an action on unordered pairs by $\ell \{u,v\}=\{\ell (u),\ell (v\left)\right\}$. A partition $\mathcal{P}$ refines a partition $\mathcal{R}$ of the same set if every block of $\mathcal{P}$ is contained in a block of $\mathcal{R}$.

Lemma 1.

The relation ${\equiv }_T$ is an equivalence relation. For its class partition $\mathcal{C}$, the group

$$K=\prod _{C\in \mathcal{C}}Sym\left(C\right),$$

acting independently on the classes, is a subgroup of $Aut\left(G\right)$. Each induced graph $G\left[C\right]$ is complete or empty. For distinct $C,D\in \mathcal{C}$, the bipartite graph $G[C,D]$ is complete bipartite or empty.

Proof. 

For distinct $u,v\in V$, condition (1) holds if and only if the transposition $\left(uv\right)$ is an automorphism of G. Indeed, this transposition fixes every unordered pair disjoint from $\{u,v\}$, fixes $\{u,v\}$ itself, and preserves adjacency on all remaining pairs precisely when u and v have the same neighbours outside $\{u,v\}$.

Reflexivity and symmetry are immediate. If $u,v,w$ are distinct and $u{\equiv }_{T}v{\equiv }_{T}w$, then

$$\left(uw\right)=\left(uv\right)\left(vw\right)\left(uv\right)\in Aut\left(G\right),$$

so $u{\equiv }_{T}w$. Cases with repeated vertices are immediate, proving transitivity.

Every transposition within a class C, extended by the identity on $V\setminus C$, is an automorphism. Such transpositions generate $Sym\left(C\right)$. The groups for different classes have disjoint supports, so their direct product K is a subgroup of $Aut\left(G\right)$.

If $\left|C\right|\ge 2$, the action of $Sym\left(C\right)$ on $\left({\displaystyle \genfrac{}{}{0pt}{}{C}{2}}\right)$ is transitive. Since adjacency is invariant under this action, $G\left[C\right]$ is complete or empty; the assertion is also true for a singleton. For distinct classes $C,D$, $Sym\left(C\right)\times Sym\left(D\right)$ is transitive on the unordered pairs with one endpoint in each class. Adjacency is therefore constant on these pairs as well. □

Define the twin quotient Q by

$$\begin{array}{cc}\hfill V\left(Q\right)& =\mathcal{C},\hfill \\ \hfill E\left(Q\right)& =\left\{\{C,D\}\in \left({\displaystyle \genfrac{}{}{0pt}{}{\mathcal{C}}{2}}\right):G[C,D]\mathrm{is}\mathrm{complete}\mathrm{bipartite}\right\}.\hfill \end{array}$$

By Lemma 1, this records exactly the adjacencies between distinct twin classes. Give each quotient vertex the colour

$$c\left(C\right)=\left(\right|C|,\tau \left(C\right)),\tau \left(C\right)=\left\{\begin{array}{cc}1,\hfill & \left|C\right|\ge 2\mathrm{and}G\left[C\right]\mathrm{is}\mathrm{complete},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(2)

Thus singleton classes all have type 0. Set

$$A=Aut(Q,c),{\mathcal{C}}_{+}=\{C\in \mathcal{C}:\tau \left(C\right)=1\},$$

where $Aut(Q,c)$ consists of the automorphisms of Q preserving c. In particular, ${\mathcal{C}}_{+}$ is invariant under A.

Remark 1

(One contraction step). Contracting the maximal twin classes once does not in general leave a twin-free graph when colours are ignored. For the three-vertex path, the two leaves form one class and the centre another; the quotient is a single edge, whose endpoints are twins but have different colours, $(2,0)$ and $(1,0)$. Quotient twins can also have equal colours: for the disjoint union of two edges, the two maximal twin classes each induce a single edge, and their quotient consists of two isolated vertices of colour $(2,1)$. Everything below concerns this single contraction step; no twin-free assumption on Q is needed.

3. The Induced Action on Twin Classes

Lemma 2.

The action on maximal twin classes defines a surjective homomorphism

$$\rho :Aut\left(G\right)⟶A,\rho \left(g\right)\left(C\right)=g\left(C\right),$$

with kernel K. Equivalently, there is a short exact sequence

$$1⟶K⟶Aut\left(G\right)\stackrel{\rho }{⟶}A⟶1.$$

(3)

Moreover, for any $\sigma \in A$, every choice of bijections $f_C:C⟶\sigma \left(C\right)$, one for each $C\in \mathcal{C}$, defines an automorphism $f={\bigcup }_{C\in \mathcal{C}}{f}_C$ of G satisfying $\rho \left(f\right)=\sigma$.

Proof. 

For $g\in Aut\left(G\right)$, the identity $N_G\left(g\left(u\right)\right)=g\left(N_G\left(u\right)\right)$ implies that $u{\equiv }_{T}v$ if and only if $g\left(u\right){\equiv }_{T}g\left(v\right)$. Consequently, g permutes the maximal twin classes. The resulting permutation preserves their sizes, internal adjacency types, and inter-class adjacencies. Hence $\rho \left(g\right)\in A$. The identity $\rho \left(gh\right)\left(C\right)=g\left(h\right(C\left)\right)$ proves that $\rho$ is a homomorphism.

An element of $ker\rho$ fixes each class setwise, so belongs to K. Conversely, Lemma 1 shows that every element of K is an automorphism fixing every class setwise. Thus $ker\rho =K$.

Let $\sigma \in A$. Colour preservation gives $\left|C\right|=\left|\sigma \right(C\left)\right|$, so the stated bijections $f_C$ exist, and their union f is a permutation of V. For distinct $u,v$ in one class C, adjacency of their images agrees with adjacency of $u,v$, since $\tau \left(\sigma \right(C\left)\right)=\tau \left(C\right)$. For $u\in C$ and $v\in D$ with $C\ne D$, Lemma 1 gives

$$\{u,v\}\in E\iff \{C,D\}\in E\left(Q\right)\iff \left\{\sigma \right(C),\sigma (D\left)\right\}\in E\left(Q\right)\iff \left\{f\right(u),f(v\left)\right\}\in E.$$

Therefore $f\in Aut\left(G\right)$ and $\rho \left(f\right)=\sigma$. This also proves surjectivity. □

Corollary 1

(The extension splits). The sequence (3) splits. After choosing an ordering of each twin class, $Aut\left(G\right)$ is isomorphic to $K⋊A$, where A acts by permuting the factors of K using these orderings. This isomorphism is noncanonical.

Proof. 

Fix a linear order on each class, writing $C=\left\{C\right(1),\dots ,C(\left|C\right|\left)\right\}$. Colour preservation gives $\left|\sigma \right(C\left)\right|=\left|C\right|$ for every $\sigma \in A$ and every $C\in \mathcal{C}$, so the assignment

$$s\left(\sigma \right)\left(C\left(i\right)\right)=\sigma \left(C\right)\left(i\right),C\in \mathcal{C},1\le i\le \left|C\right|,$$

defines a permutation of V. It is the union of the bijections $C\to \sigma \left(C\right)$ carrying $C\left(i\right)$ to $\sigma \left(C\right)\left(i\right)$, hence lies in $Aut\left(G\right)$ and satisfies $\rho \left(s\right(\sigma \left)\right)=\sigma$ by Lemma 2. For $\sigma ,\tau \in A$, both $s\left(\sigma \right)s\left(\tau \right)$ and $s\left(\sigma \tau \right)$ send $C\left(i\right)$ to $\sigma \left(\tau \right(C\left)\right)\left(i\right)$, so s is a homomorphism $A\to Aut\left(G\right)$ and therefore a section of $\rho$. The action defining the semidirect product is $\sigma ·k=s\left(\sigma \right)ks{\left(\sigma \right)}^{-1}$. Identifying each factor $Sym\left(C\right)$ with the symmetric group on $\{1,\dots ,|C\left|\right\}$ through the chosen ordering, this action is

$${(\sigma ·k)}_D=k_{{\sigma }^{-1}\left(D\right)},D\in \mathcal{C}.$$

Finally, $(k,\sigma )\mapsto ks\left(\sigma \right)$ is an isomorphism $K⋊A\to Aut\left(G\right)$: every $g\in Aut\left(G\right)$ has the unique expression $g=\left(gs{\left(\rho \left(g\right)\right)}^{-1}\right)s\left(\rho \left(g\right)\right)$, with the first factor in K. □

4. Exact Lifting of the Orbit Partitions

For $O\subseteq \mathcal{C}$ and $F\subseteq E\left(Q\right)$, define

$$\begin{array}{cc}\hfill U\left(O\right)& =\bigcup _{C\in O}C,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill X\left(F\right)& =\bigcup _{\{C,D\}\in F}\left\{\{u,v\}:u\in C,v\in D\right\}.\hfill \end{array}$$

(5)

For $O\subseteq {\mathcal{C}}_{+}$, also define

$$I\left(O\right)=\bigcup _{C\in O}\left({\displaystyle \genfrac{}{}{0pt}{}{C}{2}}\right).$$

(6)

The set in (5) does not depend on the order in which the two classes of an unordered quotient edge are written.

Theorem 1

(Twin contraction and orbit lifting). Let $G,Q,c,A,K,\rho$ be as above, with U, X and I as in (4), (5) and (6). For any subgroup $H\le A$, put ${\mathsf{\Gamma }}_H={\rho }^{-1}\left(H\right)\le Aut\left(G\right)$. Then

$${\mathcal{O}}_{{\mathsf{\Gamma }}_H}\left(V\right)=\left\{U\left(O\right):O\in {\mathcal{O}}_H\left(\mathcal{C}\right)\right\}$$

(7)

and

$${\mathcal{O}}_{{\mathsf{\Gamma }}_H}\left(E\right)=\left\{X\left(F\right):F\in {\mathcal{O}}_H\left(E\left(Q\right)\right)\right\}\cup \left\{I\left(O\right):O\in {\mathcal{O}}_H\left({\mathcal{C}}_{+}\right)\right\}.$$

(8)

Every displayed orbit block is nonempty. The two families on the right of (8) are disjoint: no edge orbit contains both an edge internal to a twin class and an edge between distinct twin classes. In particular, taking $H=A$ gives exactly the vertex and edge orbit partitions of G under its full automorphism group.

Proof. 

Lemma 2 implies that $K\le {\mathsf{\Gamma }}_H$ and that the restriction ${\rho |}_{{\mathsf{\Gamma }}_H}$ maps ${\mathsf{\Gamma }}_H$ onto H.

Let $u\in C$ and $v\in D$. If $g\left(u\right)=v$ for some $g\in {\mathsf{\Gamma }}_H$, then $\rho \left(g\right)\left(C\right)=D$, so $C,D$ lie in the same H-orbit. Conversely, if $\sigma \left(C\right)=D$ for some $\sigma \in H$, choose the bijection $f_C:C\to D$ in Lemma 2 to satisfy $f_C\left(u\right)=v$. Its extension f lies in ${\mathsf{\Gamma }}_H$ and sends u to v. Thus

$${\mathsf{\Gamma }}_{H}u={\mathsf{\Gamma }}_{H}v⟺HC=HD,$$

which proves (7).

Now consider edges $e=\{u,v\}$ and $e^{\prime }=\{u^{\prime },v^{\prime }\}$ between distinct twin classes. Write their unordered class pairs as $\{C,D\}$ and $\{C^{\prime },D^{\prime }\}$. If $g\left(e\right)=e^{\prime }$ for $g\in {\mathsf{\Gamma }}_H$, then $\rho \left(g\right)\{C,D\}=\{C^{\prime },D^{\prime }\}$. Hence their quotient edges must lie in the same H-orbit. Conversely, suppose $\sigma \{C,D\}=\{C^{\prime },D^{\prime }\}$ for some $\sigma \in H$. Lift $\sigma$ to $f\in {\mathsf{\Gamma }}_H$ using Lemma 2. The edge $f\left(e\right)$ has one endpoint in each of $C^{\prime },D^{\prime }$. Since $Sym\left(C^{\prime }\right)\times Sym\left(D^{\prime }\right)\le K$, some $k\in K$ sends these endpoints to the respective endpoints of $e^{\prime }$. Therefore $kf\left(e\right)=e^{\prime }$, with $kf\in {\mathsf{\Gamma }}_H$. It follows that the edge orbits between distinct classes are precisely the sets $X\left(F\right)$ in (8).

An edge internal to a class exists exactly when that class belongs to ${\mathcal{C}}_{+}$. Suppose $e\in \left({\displaystyle \genfrac{}{}{0pt}{}{C}{2}}\right)$ and $e^{\prime }\in \left({\displaystyle \genfrac{}{}{0pt}{}{D}{2}}\right)$ for $C,D\in {\mathcal{C}}_{+}$. If an element $g\in {\mathsf{\Gamma }}_H$ sends e to $e^{\prime }$, then $g\left(C\right)=D$, so $C,D$ lie in the same H-orbit. Conversely, choose $\sigma \in H$ with $\sigma \left(C\right)=D$ and lift it to $f\in {\mathsf{\Gamma }}_H$. The group $Sym\left(D\right)\le K$ is transitive on $\left({\displaystyle \genfrac{}{}{0pt}{}{D}{2}}\right)$, so some $k\in K$ sends $f\left(e\right)$ to $e^{\prime }$. Thus the internal edge orbits are precisely the sets $I\left(O\right)$ in (8).

Every automorphism of G preserves the twin relation. It therefore preserves the property that the endpoints of an edge lie in one twin class. The two kinds of edge orbits cannot mix. They exhaust E, and each block is nonempty by the definitions of $E\left(Q\right)$ and ${\mathcal{C}}_{+}$. This proves (8). Finally, ${\mathsf{\Gamma }}_A={\rho }^{-1}\left(A\right)=Aut\left(G\right)$, proving the last assertion. □

Corollary 2

(Exact orbit counts). The numbers of vertex and edge orbits satisfy

$$\begin{array}{cc}\hfill |{\mathcal{O}}_{Aut\left(G\right)}\left(V\right)|& =|{\mathcal{O}}_A\left(\mathcal{C}\right)|,\hfill \\ \hfill |{\mathcal{O}}_{Aut\left(G\right)}\left(E\right)|& =|{\mathcal{O}}_A\left(E\left(Q\right)\right)|+|{\mathcal{O}}_A\left({\mathcal{C}}_{+}\right)|.\hfill \end{array}$$

Proof. 

Apply Theorem 1 with $H=A$ and count its disjoint, nonempty orbit blocks. □

Corollary 3

(Refinement and bounds from partial symmetry information). For every $H\le A$, the partitions in (7)–(8) refine the full vertex and edge orbit partitions of G. In particular, for $x\in V$ or $x\in E$,

$${\mathsf{\Gamma }}_{H}x\subseteq Aut\left(G\right)x,|{\mathsf{\Gamma }}_{H}x|\le |Aut\left(G\right)x|.$$

(9)

For either $S=V$ or $S=E$, the number of blocks satisfies the reverse bound

$$|{\mathcal{O}}_{{\mathsf{\Gamma }}_H}\left(S\right)|\ge |{\mathcal{O}}_{Aut\left(G\right)}\left(S\right)|.$$

(10)

Proof. 

The inclusion ${\mathsf{\Gamma }}_H\le Aut\left(G\right)$ implies (9). Every ${\mathsf{\Gamma }}_H$-orbit is therefore contained in a full automorphism orbit, which is exactly the asserted refinement. Each block of the coarser partition is a union of one or more blocks of the finer partition, giving (10). □

Remark 2

(Interpretation of the bound). The bounds of Corollary 3 certify genuine equivalences. The reported orbit of a specified vertex or edge has size at most that of its true orbit. Thus orbit sizes are lower bounds, whereas the number of orbits is an upper bound. They become the equalities of Corollary 2 when the full colour-preserving quotient group is used. With $H=\left\{1\right\}$, the lifted vertex partition is just the twin partition.

Remark 3

(Necessary distinctions). The edge formula uses quotientedge orbits, not merely the vertex-orbit labels of their endpoints. It also includes internal clique edges, which have no quotient-edge counterpart. The colouring in (2) is essential to the general guarantee: an uncoloured quotient automorphism can exchange classes of unequal sizes or different internal types, in which case the lifting construction need not define an automorphism of G.