Abstract
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 \(\gamma_pr(G)\), is the minimum cardinality of a paired-dominating set of G. In [1], 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.
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Dorbec, P., Gravier, S. Paired-Domination in P 5-Free Graphs. Graphs and Combinatorics 24, 303–308 (2008). https://doi.org/10.1007/s00373-008-0792-x
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DOI: https://doi.org/10.1007/s00373-008-0792-x