Tyshkevich's graph decomposition and the distinguishing numbers of unigraphs

Abstract

A c-labeling $\varphi :V\left(G\right)\to \{1,2,\dots ,c\}$ of graph G is distinguishing if, for every non-trivial automorphism π of G, there is some vertex v so that $\varphi \left(v\right)\ne \varphi \left(\pi \right(v\left)\right)$. The distinguishing number of G, $D\left(G\right)$, is the smallest c such that G has a distinguishing c-labeling.

We consider a compact version of Tyshkevich's graph decomposition theorem where trivial components are maximally combined to form a complete graph or a graph of isolated vertices. Suppose the compact canonical decomposition of G is $G_k\circ G_{k-1}\circ \cdots \circ G_1\circ G_0$. We prove that ϕ is a distinguishing labeling of G if and only if ϕ is a distinguishing labeling of $G_i$ when restricted to $V\left(G_i\right)$ for $i=0,\dots ,k$. Thus, $D\left(G\right)=\max \left\{D\right(G_i),i=0,\dots ,k\}$. We then present an algorithm that computes the distinguishing number of a unigraph in linear time.