Algorithmic aspects of semitotal domination in graphs

Abstract

For a graph G = (V, E), a set D ⊆ V is called a semitotal dominating set of G if D is a dominating set of G, and every vertex in D is within distance 2 of another vertex of D. The Minimum Semitotal Domination problem is to find a semitotal dominating set of minimum cardinality. Given a graph G and a positive integer k, the Semitotal Domination Decision problem is to decide whether G has a semitotal dominating set of cardinality at most k. The Semitotal Domination Decision problem is known to be NP-complete for general graphs. In this paper, we show that the Semitotal Domination Decision problem remains NP-complete for planar graphs, split graphs and chordal bipartite graphs. We give a polynomial time algorithm to solve the Minimum Semitotal Domination problem in interval graphs. We show that the Minimum Semitotal Domination problem in a graph with maximum degree ∆ admits an approximation algorithm that achieves the approximation ratio of 2 + 3 ln(∆ + 1), showing that the problem is in the class log-APX. We also show that the Minimum Semitotal Domination problem cannot be approximated within (1 − ǫ) ln |V | for any ǫ > 0 unless NP ⊆ DTIME (|V | O(log log |V |) ). Finally, we prove that the Minimum Semitotal Domination problem is APX-complete for bipartite graphs with maximum degree 4.

Semitotal Domination: New hardness results and a polynomial-time algorithm for graphs of bounded mim-width

“…Note that the same inapproximability result holds for DOMINATING SET, TOTAL DOMINATING SET and CONNECTED DOMINATING SET [14]. On the other hand, using the natural greedy algorithm for SET COVER, Henning and Pandey [31] showed that SEMITOTAL DOMINATING SET is in APX for graphs with bounded degree: it admits a 2 + 3 ln(∆ + 1) approximation algorithm for graphs with maximum degree ∆. Moreover, they showed that it is APX-complete for bipartite graphs with maximum degree 4 and asked whether the same holds for subcubic graphs.…”

“…Henning and Pandey [31] showed that SEMITOTAL DOMINATING SET has the same approximation hardness as the well-known SET COVER: SEMITOTAL DOMINATING SET is not approximable within (1 − ε) ln n, for any ε > 0, unless NP ⊆ DTIME(n O(log log n) ). Note that the same inapproximability result holds for DOMINATING SET, TOTAL DOMINATING SET and CONNECTED DOMINATING SET [14].…”

“…Finally, suppose H is the disjoint union of paths. If H contains at least two paths on at least 2 vertices, then H contains 2K 2 and the problem is NP-complete since it is NP-complete when restricted to split graphs [31] (see also Theorem 14). The same conclusion holds if H contains a path on at least 5 vertices.…”

“…Graph class DOMINATING SET SEMITOTAL DOMINATING SET TOTAL DOMINATING SET bipartite NP-c [7] NP-c [31] NP-c [49] line graph of bipartite NP-c [37] NP-c Since γ(G) ≤ γ t2 (G) ≤ γ t (G), for any graph G with no isolated vertex, it is natural to ask for which graphs equalities hold. In Section 8, we show that the graphs attaining either of the two equalities are unlikely to have a polynomial characterisation.…”

Complexity and Algorithms for Semipaired Domination in Graphs

“…We show that the decision version of the MINIMUM PAIRED DOMINATION problem is NP-complete for GP4 graphs, but the MINIMUM SEMIPAIRED DOMINATION problem is easily solvable for GP4 graphs. The class of GP4 graphs was introduced by Henning and Pandey in [15]. Below we recall the definition of GP4 graphs.…”