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.

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A set D ⊆ V of a graph G = (V, E) is called an open neighborhood locating-dominating set (OLD-set) ifGiven a graph G = (V, E), the MIN OLD-SET problem is to find an OLD-set of minimum cardinality. The cardinality of a minimum OLD-set of G is called the open neighborhood location-domination number of G, and is denoted by γ old (G). Given a graph G and a positive integer k, the DECIDE OLD-SET problem is to decide whether G has an OLDset of cardinality at most k. The DECIDE OLD-SET problem is known to be NP-complete for bipartite graphs. In this paper, we strengthen this NP-complete result by showing that the DECIDE OLD-SET problem remains NP-complete for perfect elimination bipartite graphs, a subclass of bipartite graphs. Then, we show that the MIN OLD-SET problem can be solved in polynomial time in chain graphs, a subclass of perfect elimination bipartite graphs. We show that for a graph G, γ old (G) ≥ 2n Δ(G)+2 , where n denotes the number of vertices in G, and Δ(G) denotes the maximum degree of G. As a consequence we obtain a Δ(G)+2 2 -approximation algorithm for the MIN OLD-SET problem. Finally, we prove that the MIN OLD-SET problem is APX-complete for chordal graphs with maximum degree 4.

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