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

• Published in: Journal of applied mathematics & computing Vol. 66; no. 1-2; pp. 733 - 747
• Main Authors: Pradhan, D., Pal, Saikat
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.06.2021; Springer Nature B.V
• Subjects: Algorithms; Apexes; Applied mathematics; Computational Mathematics and Numerical Analysis; Graph theory; Graphs; Mathematical and Computational Engineering; Mathematics; Mathematics and Statistics; Mathematics of Computing; Original Research; Theory of Computation; Domination; Interval graphs; 68Q25; Total domination; Polynomial time algorithm; Semitotal domination; 05C69
• ISSN: 1598-5865; 1865-2085
• DOI: 10.1007/s12190-020-01459-9

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 ) .