Edge-apexing in hereditary classes of graphs

TL;DR

This work studies the edge-apex transformation on hereditary graph classes, proving that finiteness of forbidden induced subgraphs is preserved under epex and providing a vertex-bound framework. It specializes to cographs, establishing a characterization of edge-apex cographs via forbidden induced subgraphs and performing a computer-assisted enumeration of all exclusions up to eight vertices. The results include a tight 5–8 vertex bound and a complete list of forbidden subgraphs for edge-apex cographs, obtained with SageMath. The findings have implications for structural graph theory and algorithms for recognizing near-cograph graphs.

Abstract

A class $\mathcal{G}$ of graphs is called hereditary if it is closed under taking induced subgraphs. We denote by $G^{epex}$ the class of graphs that are at most one edge away from being in $\mathcal{G}$. We note that $G^{epex}$ is hereditary and prove that if a hereditary class $\mathcal{G}$ has finitely many forbidden induced subgraphs, then so does $G^{epex}$. The hereditary class of cographs consists of all graphs $G$ that can be generated from $K_1$ using complementation and disjoint union. Cographs are precisely the graphs that do not have the $4$-vertex path as an induced subgraph. For the class of edge-apex cographs our main result bounds the order of such forbidden induced subgraphs by 8 and finds all of them by computer search.

Figures (4)

• Figure 1: The $5$-vertex forbidden induced subgraphs for edge-apex cographs.
• Figure 2: The $6$-vertex forbidden induced subgraphs for edge-apex cographs.
• Figure 3: The $7$-vertex forbidden induced subgraphs for edge-apex cographs.
• Figure 4: The three $8$-vertex forbidden induced subgraphs for edge-apex cographs.

Theorems & Definitions (17)

• Theorem 1.1: corneil
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Lemma 3.1
• proof