Abstract:
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
