Semipaired Domination in Some Subclasses of Chordal Graphs

TL;DR

This paper investigates the computational complexity of the semipaired domination problem in various graph classes, proving NP-completeness in split graphs, and providing efficient algorithms for block graphs, along with complexity results for degree-restricted graphs.

Contribution

It establishes NP-completeness of the problem in split graphs, offers a linear-time algorithm for block graphs, and proves APX-completeness for graphs with maximum degree 3.

Findings

NP-complete for split graphs

Linear-time algorithm for block graphs

APX-complete for degree 3 graphs

Abstract

A dominating set $D$ of a graph $G$ without isolated vertices is called semipaired dominating set if $D$ can be partitioned into $2$-element subsets such that the vertices in each set are at distance at most $2$. The semipaired domination number, denoted by $\gamma_{pr2}(G)$ is the minimum cardinality of a semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices, the \textsc{Minimum Semipaired Domination} problem is to find a semipaired dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of the \textsc{Minimum Semipaired Domination} problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the \textsc{Minimum Semipaired Domination} problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the \textsc{Minimum Semipaired Domination} problem is APX-complete for graphs with maximum degree $3$.