A linear-time algorithm for semitotal domination 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.