Tree independence number I. (Even hole, diamond, pyramid)-free graphs

TL;DR

This work establishes a bounded tree-independence number for the class $\mathcal{C}$ of ($C_4$, diamond, theta, pyramid, prism, even wheel)-free graphs by developing a central-bag framework anchored in separations and hub divisions. The authors connect $\mathop{\mathrm{tree-\alpha}}$ to the clique-cover based parameter $\mathop{\mathrm{tree-\overline{\chi}}}$, and show that for $\mathcal{C}$, central bags admit small balanced separators in terms of clique covers, enabling a global bound via extending separators from bags to the whole graph. Key steps include trisimplicial elimination orderings, wheel/3PC avoidance to control structure, and a sequence of lemmas relating separators, anchors, and hub divisions. The main result is that there exists $\Gamma$ with $\mathop{\mathrm{tree-\alpha}}(G) \le \Gamma$ for all $G \in \mathcal{C}$, yielding a polynomial-time algorithm for Maximum Weight Independent Set in this class and corroborating a conjecture linking tree-\alpha and bounded treewidth in hereditary classes. The approach provides a framework potentially extensible to broader even-hole-free families and highlights the central-bag technique as a robust tool for algorithmic graph problems in structured hereditary classes.

Abstract

The tree-independence number tree-$α$, first defined and studied by Dallard, Milanič and Štorgel, is a variant of treewidth tailored to solving the maximum independent set problem. Over a series of papers, Abrishami et al. developed the so-called central bag method to study induced obstructions to bounded treewidth. Among others, they showed that, in a certain superclass $\mathcal C$ of (even hole, diamond, pyramid)-free graphs, treewidth is bounded by a function of the clique number. In this paper, we relax the bounded clique number assumption, and show that $\mathcal C$ has bounded tree-$α$. Via existing results, this yields a polynomial time algorithm for the maximum independent set problem in this class. Our result also corroborates, for this class of graphs, a conjecture of Dallard, Milanič and Štorgel that in a hereditary graph class, tree-$α$ is bounded if and only if the treewidth is bounded by a function of the clique number.

Theorems & Definitions (45)

• Conjecture 1.1: dms2
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Conjecture 1.6
• Conjecture 1.7
• Lemma 2.1: wagon
• Lemma 2.2
• Remark 2.3