Tyshkevich's Graph Decomposition and the Distinguishing Numbers of Unigraphs

Abstract

A c-labeling φ: V(G) → \1, 2, , c \ of graph G is distinguishing if, for every non-trivial automorphism π of G, there is some vertex v so that φ(v) ≠ φ(π(v)). The distinguishing number of G, D(G), 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 Gk Gk-1 ·s G1 G0. We prove that φ is a distinguishing labeling of G if and only if φ is a distinguishing labeling of Gi when restricted to V(Gi) for i = 0, , k. Thus, D(G) = \D(Gi), i = 0, , k \. We then present an algorithm that computes the distinguishing number of a unigraph in linear time.