Semipaired domination in some subclasses of chordal graphs

A dominating set D of a graph G without isolated vertices is called semipaired dominating set if D can be partitioned into 2-element subsets such that the vertices in each set are at distance at most 2. The semipaired domination number, denoted by γpr2(G) is the minimum cardinality of a semipaired dominating set of G. Given a graph G with no isolated vertices, the MINIMUM SEMIPAIRED DOMINATION problem is to find a semipaired dominating set of G of cardinality γpr2(G). The decision version of the MINIMUM SEMIPAIRED DOMINATION problem is already known to be NP-complete for chordal graphs, an important graph class. In this paper, we show that the decision version of the MINIMUM SEMIPAIRED DOMINATION problem remains NP-complete for split graphs, a subclass of chordal graphs. On the positive side, we propose a linear-time algorithm to compute a minimum cardinality semipaired dominating set of block graphs. In addition, we prove that the MINIMUM SEMIPAIRED DOMINATION problem is APX-complete for graphs with maximum degree 3.