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

• Volume: 32, Issue: 3, page 473-485
• ISSN: 2083-5892

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## Abstract

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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.

## How to cite

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Oliver Schaudt. "Paired- and induced paired-domination in {E,net}-free graphs." Discussiones Mathematicae Graph Theory 32.3 (2012): 473-485. <http://eudml.org/doc/271009>.

@article{OliverSchaudt2012,
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 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. },
author = {Oliver Schaudt},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {domination; paired-domination; induced paired-domination; induced matchings; \{E,net\}-free graphs; -free graphs},
language = {eng},
number = {3},
pages = {473-485},
title = {Paired- and induced paired-domination in \{E,net\}-free graphs},
url = {http://eudml.org/doc/271009},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Oliver Schaudt
TI - Paired- and induced paired-domination in {E,net}-free graphs
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 3
SP - 473
EP - 485
AB - 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.
LA - eng
KW - domination; paired-domination; induced paired-domination; induced matchings; {E,net}-free graphs; -free graphs
UR - http://eudml.org/doc/271009
ER -

## References

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