  A graph $G$ is a {\em chordal-$k$-generalized split graph} if $G$ is chordal
and there is a clique $Q$ in $G$ such that every connected component in $G[V
\setminus Q]$ has at most $k$ vertices. Thus, chordal-$1$-generalized split
graphs are exactly the split graphs.
  We characterize chordal-$k$-generalized split graphs by forbidden induced
subgraphs. Moreover, we characterize a very special case of
chordal-$2$-generalized split graphs for which the Efficient Domination problem
is \NP-complete.
