Apex Graphs and Cographs

TL;DR

This paper characterizes the forbidden induced subgraphs for the class of apex cographs, which are graphs that become cographs upon removal of a single vertex, extending understanding of hereditary graph classes.

Contribution

It identifies all forbidden induced subgraphs for apex cographs, providing a complete structural characterization of this class.

Findings

Apex cographs are characterized by a finite set of forbidden induced subgraphs.

The paper extends the theory of hereditary graph classes with finitely many forbidden subgraphs.

It demonstrates that the property of having finitely many forbidden subgraphs is preserved under apex operations.

Abstract

A class $\mathcal{G}$ of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by $\mathcal{G}^\mathrm{apex}$ the class of graphs $G$ that contain a vertex $v$ such that $G-v$ is in $\mathcal{G}$. We prove that if a hereditary class $\mathcal{G}$ has finitely many forbidden induced subgraphs, then so does $\mathcal{G}^\mathrm{apex}$. The hereditary class of cographs consists of all graphs $G$ that can be generated from $K_1$ using complementation and disjoint union. A graph is an apex cograph if it contains a vertex whose deletion results in a cograph. Cographs are precisely the graphs that do not have the $4$-vertex path as an induced subgraph. Our main result finds all such forbidden induced subgraphs for the class of apex cographs.