Hereditary classes of graphs and matroids with finitely many exclusions
Jagdeep Singh, Vaidy Sivaraman
TL;DR
The article extends the study of hereditary graph classes by proving that the edge-add operation preserves finiteness of forbidden induced subgraphs: if a hereditary class $\mathcal{G}$ has finitely many forbidden subgraphs, then $\mathcal{G}^{\mathrm{add}}$ does as well, with further implications for classes formed by bounded edge/vertex modifications. It provides explicit finite forbidden subgraph lists for edge-add split graphs and edge-add threshold graphs (and, via complementation, for their edge-apex counterparts), and introduces $(p,q)$-edge-split graphs and edge-add chordal graphs with corresponding finiteness results. The paper also establishes matching results in the binary/ternary matroid setting, giving rank-based bounds for forbidden flats in $\mathcal{M}^{\mathrm{add}}$ and developing a projective-geometry representation via 2-colorings to prove these bounds. Collectively, the work strengthens Gyárfás-style finiteness results across graph and matroid settings and provides practical templates (lists of forbidden subgraphs and structural decompositions) for recognizing near-membership in these augmented hereditary classes.
Abstract
A class $\mathcal{G}$ of graphs closed under taking induced subgraphs is called hereditary. We denote by $\mathcal{G}^\mathrm{add}$ the class of graphs at most one edge addition away from $\mathcal{G}$, and by $\mathcal{G}^\mathrm{epex}$ the class of graphs at most one edge deletion away. We previously showed that if $\mathcal{G}$ has finitely many forbidden induced subgraphs, then so does the hereditary class $\mathcal{G}^\mathrm{epex}$. In this paper, we prove the corresponding result for $\mathcal{G}^\mathrm{add}$. Consequently, we show that the class of graphs within a fixed number of vertex deletions, edge deletions, and edge additions from $\mathcal{G}$ also has finitely many forbidden induced subgraphs provided that $\mathcal{G}$ does. We also present the binary and ternary matroid analogue of this result. Additionally, if $\mathcal{G}$ is closed under complementation, the forbidden induced subgraphs of $\mathcal{G}^\mathrm{add}$ and $\mathcal{G}^\mathrm{epex}$ are complements of each other. We provide explicit lists of the forbidden induced subgraphs for $\mathcal{G}^\mathrm{add}$, and consequently for $\mathcal{G}^\mathrm{epex}$, when $\mathcal{G}$ is the class of split graphs, cographs, or threshold graphs. Following Gyárfás's framework, we introduce $(p,q)$-edge split graphs, analogous to his $(p,q)$-split graphs, and prove they have finitely many forbidden induced subgraphs.