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Paired-domination in \(P_{5}\)-free graphs. (English) Zbl 1193.05123

Summary: A set \(S\) of vertices in a graph \(G\) is a paired-dominating set of \(G\) if every vertex of \(G\) is adjacent to some vertex in \(S\) and if the subgraph induced by \(S\) contains a perfect matching. The paired-domination number of \(G\), denoted by \(\gamma_{\text{pr}}(G)\), is the minimum cardinality of a paired-dominating set of \(G\). In [P. Dorbec, S. Gravier, and M.A. Henning, “Paired-domination in generalized claw-free graphs”, J. Comb. Optim. 14, No. 1, 1–7 (2007; Zbl 1180.05088)], the authors gave tight bounds for paired-dominating sets of generalized claw-free graphs. Yet, the critical cases are not claws but subdivided stars. We here give a bound for graphs containing no induced \(P_{5}\), which seems to be the critical case.

MSC:

05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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