A linear-time algorithm for semitotal domination in strongly chordal graphs

TL;DR

This paper presents a linear-time algorithm for solving the minimum semitotal domination problem specifically in strongly chordal graphs, a class where the problem's complexity was previously unresolved.

Contribution

The paper introduces the first linear-time algorithm for the minimum semitotal domination problem in strongly chordal graphs, resolving an open complexity question.

Findings

The problem is solvable in linear time for strongly chordal graphs.

The algorithm improves upon previous polynomial-time solutions for this class.

It confirms the problem's complexity status as tractable in strongly chordal graphs.

Abstract

In a graph $G=(V,E)$ with no isolated vertex, a dominating set $D \subseteq V$, is called a semitotal dominating set if for every vertex $u \in D$ there is another vertex $v \in D$, such that distance between $u$ and $v$ is at most two in $G$. Given a graph $G=(V,E)$ without isolated vertices, the Minimum Semitotal Domination problem is to find a minimum cardinality semitotal dominating set of $G$. The semitotal domination number, denoted by $\gamma_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. The decision version of the problem remains NP-complete even when restricted to chordal graphs, chordal bipartite graphs, and planar graphs. Galby et al. in [6] proved that the problem can be solved in polynomial time for bounded MIM-width graphs which includes many well known graph classes, but left the complexity of the problem in strongly chordal graphs unresolved. Henning and Pandey in [20] also asked to resolve the complexity status of the problem in strongly chordal graphs. In this paper, we resolve the complexity of the problem in strongly chordal graphs by designing a linear-time algorithm for the problem.