An $$O(n+m)$$ time algorithm for computing a minimum semitotal dominating set in an interval graph

Abstract

Let $$G=(V,E)$$ be a graph without isolated vertices. A set $$D\subseteq V$$ is said to be a dominating set of G if for every vertex $$v\in V\setminus D$$ , there exists a vertex $$u\in D$$ such that $$uv\in E$$ . A set $$D\subseteq V$$ is called a semitotal dominating set of G if D is a dominating set and every vertex in D is within distance 2 from another vertex of D. For a given graph G, the semitotal domination problem is to find a semitotal dominating set of G with minimum cardinality. The decision version of the semitotal domination problem is shown to be NP-complete for chordal graphs and bipartite graphs. Henning and Pandey (Theor Comput Sci 766:46–57, 2019) proposed an $$O(n^2)$$ time algorithm for computing a minimum semitotal dominating set in interval graphs. In this paper, we show that for a given interval graph $$G=(V,E)$$ , a minimum semitotal dominating set of G can be computed in $$O(n+m)$$ time, where $$n=|V|$$ and $$m=|E|$$ . This improves the complexity of the semitotal domination problem for interval graphs from $$O(n^2)$$ to $$O(n+m)$$ .