Tyshkevich's Graph Decomposition and the Distinguishing Numbers of Unigraphs
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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 G_k∘ G_k-1∘⋯∘ G_1 ∘ 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(G_i) for i = 0, , k. Thus, D(G) = max{D(G_i), i = 0, , k }. We then present an algorithm that computes the distinguishing number of a unigraph in linear time.
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