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Paired- and induced paired-domination in {E,net}-free graphs

100%

Schaudt O.

Discussiones Mathematicae Graph Theory

2012

tom 32

nr 3

473-485

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.

Total domination versus paired domination

100%

Schaudt O.

Discussiones Mathematicae Graph Theory

2012

tom 32

nr 3

435-447

A dominating set of a graph G is a vertex subset that any vertex of G either belongs to or is adjacent to. A total dominating set is a dominating set whose induced subgraph does not contain isolated vertices. The minimal size of a total dominating set, the total domination number, is denoted by γₜ. The maximal size of an inclusionwise minimal total dominating set, the upper total domination number, is denoted by Γₜ. A paired dominating set is a dominating set whose induced subgraph has a perfect matching. The minimal size of a paired dominating set, the paired domination number, is denoted by γₚ. The maximal size of an inclusionwise minimal paired dominating set, the upper paired domination number, is denoted by Γₚ. In this paper we prove several results on the ratio of these four parameters: For each r ≥ 2 we prove the sharp bound γₚ/γₜ ≤ 2 - 2/r for $K_{1,r}$-free graphs. As a consequence, we obtain the sharp bound γₚ/γₜ ≤ 2 - 2/(Δ+1), where Δ is the maximum degree. We also show for each r ≥ 2 that ${C₅,T_r}$-free graphs fulfill the sharp bound γₚ/γₜ ≤ 2 - 2/r, where $T_r$ is obtained from $K_{1,r}$ by subdividing each edge exactly once. We show that all of these bounds also hold for the ratio Γₚ/Γₜ. Further, we prove that a graph hereditarily has an induced paired dominating set if and only if γₚ ≤ Γₜ holds for any induced subgraph. We also give a finite forbidden subgraph characterization for this condition. We exactly determine the maximal value of the ratio γₚ/Γₜ taken over the induced subgraphs of a graph. As a consequence, we prove for each r ≥ 3 the sharp bound γₚ/Γₜ ≤ 2 - 2/r for graphs that do not contain the corona of $K_{1,r}$ as subgraph. In particular, we obtain the sharp bound γₚ/Γₜ ≤ 2 - 2/Δ.

Ograniczanie wyników

2 Discussiones Mathematicae Graph Theory

2 Schaudt O.

2 2012

Wyniki wyszukiwania

Paired- and induced paired-domination in {E,net}-free graphs

Total domination versus paired domination