Tree-alpha and excluding finitely many graphs

TL;DR

This paper characterizes hereditary graph classes with bounded tree-alpha based on forbidden subgraphs, resolving two conjectures related to graph exclusion and boundedness properties.

Contribution

It proves that such classes are characterized by excluding specific graphs, confirming two conjectures about hereditary graph classes and their bounded tree-alpha.

Findings

Hereditary classes with finitely many forbidden induced subgraphs have bounded tree-alpha iff they are (tw,ω)-bounded.

Such classes exclude a complete bipartite graph, a forest with at most three leaves per component, and the line graph of such a forest.

Resolved two conjectures regarding bounded tree-alpha and hereditary graph classes excluding certain subgraphs.

Abstract

We prove that a hereditary graph class $\mathcal{G}$ defined by finitely many excluded induced subgraphs has bounded tree-$\alpha$ if and only if it is "$(\mathrm{tw},\omega)$-bounded" (that is, for all $t\in \mathbb N$, the class of all $K_t$-free graphs in $\mathcal{G}$ has bounded treewidth). Equivalently, $\mathcal{G}$ has bounded tree-$\alpha$ if and only if it excludes a complete bipartite graph, a forest whose components each have at most three leaves, and the line graph of such a forest. This resolves two conjectures of Dallard, Krnc, Kwon, Milani\v{c}, Munaro, \v{S}torgel, and Wiederrecht: the above, and a weaker one that for all $a,b\in \mathbb N$, every hereditary class that excludes $K_{a,a}$ and the $b$-vertex path has bounded tree-$\alpha$. The latter was already open even for $(a,b)\in \{(2,7),(3,5)\}$, and only recently proved for $(a,b)=(2,6)$.