On Chordal-$k$-Generalized Split Graphs

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.