## 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.)

### Keywords:

paired-domination; $$P_{5}$$-free; subdivided star; bounds

### Citations:

Zbl 0904.05052; Zbl 1180.05088
Full Text:

### References:

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