Paired and semipaired domination in triangulations

TL;DR

This paper investigates paired and semipaired domination numbers in near-triangulation graphs, establishing upper bounds for these parameters based on the number of vertices, with specific bounds proven for different cases.

Contribution

The paper provides new upper bounds for the paired and semipaared domination numbers in near-triangulation graphs, extending understanding of domination parameters in planar graphs.

Findings

Paired domination number $ abla_{pr}(G) leq 2 loor{rac{n}{4}}$ for near-triangulations.

Semipaired domination number $ abla_{pr2}(G) leq loor{rac{2n}{5}}$ with some exceptions.

Bounds are tight or nearly tight for large classes of near-triangulation graphs.

Abstract

A dominating set of a graph $G$ is a subset $D$ of vertices such that every vertex not in $D$ is adjacent to at least one vertex in $D$. A dominating set $D$ is paired if the subgraph induced by its vertices has a perfect matching, and semipaired if every vertex in $D$ is paired with exactly one other vertex in $D$ that is within distance 2 from it. The paired domination number, denoted by $\gamma_{pr}(G)$, is the minimum cardinality of a paired dominating set of $G$, and the semipaired domination number, denoted by $\gamma_{pr2}(G)$, is the minimum cardinality of a semipaired dominating set of $G$. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that $\gamma_{pr}(G) \le 2 \lfloor \frac{n}{4} \rfloor$ for any near-triangulation $G$ of order $n\ge 4$, and that with some exceptions, $\gamma_{pr2}(G) \le \lfloor \frac{2n}{5} \rfloor$ for any near-triangulation $G$ of order $n\ge 5$.