Common domination perfect graphs
Abstract
A dominating set in a graph G is a set S of vertices such that every vertex that does not belong to S is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality of a dominating set of G. The common independence number αc(G) of G is the greatest integer r such that every vertex of G belongs to some independent set of cardinality at least~r. The common independence number is squeezed between the independent domination number i(G) and the independence number α(G) of G, that is, γ(G) i(G) αc(G) α(G). A graph G is domination perfect if γ(H) = i(H) for every induced subgraph H of G. We define a graph G as common domination perfect if γ(H) = αc(H) for every induced subgraph H of G. We provide a characterization of common domination perfect graphs in terms of ten forbidden induced subgraphs.
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