Forbidden induced subgraph characterization of circle graphs within split graphs

TL;DR

This paper characterizes circle graphs within split graphs by identifying minimal forbidden induced subgraphs, providing a structural understanding of these graphs.

Contribution

It offers the first known forbidden induced subgraph characterization of circle graphs within split graphs.

Findings

Identifies minimal forbidden induced subgraphs for circle graphs in split graphs

Provides a structural characterization of circle graphs in this class

Advances understanding of graph intersection properties

Abstract

A graph is circle if its vertices are in correspondence with a family of chords in a circle in such a way that every two distinct vertices are adjacent if and only if the corresponding chords have nonempty intersection. Even though there are diverse characterizations of circle graphs, a structural characterization by minimal forbidden induced subgraphs for the entire class of circle graphs is not known, not even restricted to split graphs (which are the graphs whose vertex set can be partitioned into a clique and a stable set). In this work, we give a characterization by minimal forbidden induced subgraphs of circle graphs, restricted to split graphs.