An O(n+m) time algorithm for computing a minimum semitotal dominating set in an interval graph - Pradhan, D.; Pal, Saikat (Journal of applied mathematics & computing, 2021)
Let G = ( V , E ) be a graph without isolated vertices. A set D ⊆ V is said to be a dominating set of G if for every vertex v ∈ V \ D , there exists a vertex u ∈ D such that u v ∈ E . A set D ⊆ 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 ) .