# On Chordal-$k$-Generalized Split Graphs

## TL;DR

This paper characterizes chordal-$k$-generalized split graphs through forbidden subgraphs and explores the computational complexity of the Efficient Domination problem in a special case.

## Contribution

It provides a forbidden subgraph characterization for chordal-$k$-generalized split graphs and analyzes the NP-completeness of the Efficient Domination problem in a specific case.

## Key findings

- Chordal-$k$-generalized split graphs are characterized by forbidden induced subgraphs.
- The paper identifies a special case of chordal-$2$-generalized split graphs where the Efficient Domination problem is NP-complete.
- Split graphs are exactly the chordal-$1$-generalized split graphs.

## Abstract

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.