Abstract
Let β(G) and Γ(G) be the independence number and the upper domination number of a graph G, respectively. A graph G is called Γ-perfect if β(H) = Γ(H), for every induced subgraph H of G. The class of Γ-perfect graphs generalizes such well-known classes of graphs as strongly perfect graphs, absobantly perfect graphs, and circular arc graphs. In this article, we present a characterization of Γ-perfect graphs in terms of forbidden semi-induced subgraphs. Key roles in the characterization are played by the odd prism and the even Möbius ladder, where the prism and the Möbius ladder are well-known 3-regular graphs [2]. Using the semi-induced subgraph characterization, we obtain a characterization of K1,3-free Γ-perfect graphs in terms of forbidden induced subgraphs. © 1999 John Wiley & Sons, Inc. J Graph Theory 31:29–49, 1999
Highlights
All graphs will be finite and undirected without multiple edges
Recall that a graph G is called strongly perfect if every induced subgraph H of G has a stable transversal, where a stable transversal S of H is a vertex subset of H such that |S ∩ C| = 1 for any maximal clique C of H
Hammer and Maffray [15] defined a graph G to be absorbantly perfect if every induced subgraph H of G contains a minimal dominating set that meets all maximal cliques of H
Summary
All graphs will be finite and undirected without multiple edges. Unless otherwise stated, all graphs have no loops. Definition 2 A graph G is called domination perfect (γ-perfect) if γ(H) = i(H), for every induced subgraph H of G. Definition 3 A graph G is called upper domination perfect (Γ-perfect) if β(H) = Γ(H), for every induced subgraph H of G. Definition 4 A graph G is called upper irredundance perfect (IR-perfect) if Γ(H) = IR(H), for every induced subgraph H of G. Recall that a graph G is called strongly perfect if every induced subgraph H of G has a stable transversal, where a stable transversal S of H is a vertex subset of H such that |S ∩ C| = 1 for any maximal clique C of H. Hammer and Maffray [15] defined a graph G to be absorbantly perfect if every induced subgraph H of G contains a minimal dominating set that meets all maximal cliques of H. Notice here that K1,3-free graphs are γ-perfect [1], and that K1,3-free ir-perfect graphs were characterized by Favaron [10], who found a sufficient condition for a K1,3-free graph to be IR-perfect
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