A dominating set of a graph is a vertex subset that any vertex belongs to or is adjacent to. Among the many well-studied variants of domination are the so-called paired-dominating sets. A paired-dominating set is a dominating set whose induced subgraph has a perfect matching. In this paper, we continue their study. We focus on graphs that do not contain the net-graph (obtained by attaching a pendant vertex to each vertex of the triangle) or the E-graph (obtained by attaching a pendant vertex to each vertex of the path on three vertices) as induced subgraphs. This graph class is a natural generalization of {claw, net}-free graphs, which are intensively studied with respect to their nice properties concerning domination and hamiltonicity. We show that any connected {E, net}-free graph has a paired-dominating set that, roughly, contains at most half of the vertices of the graph. This bound is a significant improvement to the known general bounds. Further, we show that any {E, net, C₅}-free graph has an induced paired-dominating set, that is a paired-dominating set that forms an induced matching, and that such set can be chosen to be a minimum paired-dominating set. We use these results to obtain a new characterization of {E, net, C₅}-free graphs in terms of the hereditary existence of induced paired-dominating sets. Finally, we show that the induced matching formed by an induced paired-dominating set in a {E, net, C₅}-free graph can be chosen to have at most two times the size of the smallest maximal induced matching possible.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1617,
author = {Oliver Schaudt},
title = {Paired- and induced paired-domination in {E,net}-free graphs},
journal = {Discussiones Mathematicae Graph Theory},
volume = {32},
year = {2012},
pages = {473-485},
zbl = {1257.05122},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1617}
}
Oliver Schaudt. Paired- and induced paired-domination in {E,net}-free graphs. Discussiones Mathematicae Graph Theory, Tome 32 (2012) pp. 473-485. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1617/
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