  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 $\gamma_{pr2}(G)$ is the minimum cardinality of a
semipaired dominating set of $G$. Given a graph $G$ with no isolated vertices,
the \textsc{Minimum Semipaired Domination} problem is to find a semipaired
dominating set of $G$ of cardinality $\gamma_{pr2}(G)$. The decision version of
the \textsc{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 \textsc{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 \textsc{Minimum Semipaired Domination} problem is APX-complete for
graphs with maximum degree $3$.
