Paired- and induced paired-domination in (E,net)-free graphs

Abstract

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 paireddominating set, that is a paired-dominating set that forms an induced matching, and that such set can be chosen to be a minimum paireddominating sets. 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.

Upper paired domination versus upper domination

“…For further information about lower bounds for paired domination number, the reader is referred to a comprehensive survey by (Desormeaux et al (2020)). Furthermore, studies on the relationship between paired domination number and other domination variants such as total domination number (Shan et al (2004), Schaudt (2012b), Cyman et al (2018)), induced paired-domination (Schaudt (2012a)), and double domination (Dorbec et al (2014)) form the third category.…”