Induced-Minor-Closed Classes have Linear, Square-Root, or Sub-Polynomial Tree-Independence

2026-07-13

AI Summary

This study investigates upper bounds on the tree-independence number of graph classes closed under induced subgraphs and their algorithmic implications. By leveraging graph decomposition theory, induced subgraph analysis, and extremal combinatorial methods, it is shown that any graph excluding a complete bipartite graph $K_{t,t}$ or a grid graph $W_{t\times t}$ as an induced subgraph has subpolynomial tree-independence number. This result yields the first trichotomy for such graph classes based on their tree-independence number—linear, square-root, or subpolynomial—and partially resolves a conjecture by Chudnovsky et al. regarding polylogarithmic bounds. The derived bound of $O(2^{O((\log n)^{1-\varepsilon})})$ leads to a universal algorithm running in time $2^{n^{o(1)}}$, applicable to a broad range of graph optimization problems.

Abstract

An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. A tree decomposition of $G$ is a pair $(T, χ)$ where $T$ is a tree and $χ: V(T) \rightarrow 2^{V(G)}$ is a function satisfying two axioms: for every edge $uv \in E(G)$ there is an $x \in V(T)$ such that $\{u,v\} \subseteq χ(x)$, and for every vertex $u \in V(G)$ the set $\{x \in V(T) | u \in χ(x)\}$ induces a non-empty and connected subtree of $T$. The sets $χ(x)$ for $x \in V(T)$ are called the bags of the tree decomposition. The tree-independence number of $G$ is the minimum taken over all tree decompositions of $G$ of the maximum size of an independent set of the graph induced by a bag of the decomposition. A graph $H$ is an induced minor of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by vertex deletions and edge contractions. We prove that for every $t\in\mathbb{N}$ there exists an $ε> 0$ such that every graph $G$ either contains the complete bipartite graph $K_{t,t}$ or the wall $W_{t\times t}$ as an induced minor, or has tree-independence at most $O(2^{O((\log n)^{1-ε})})$. This leads to algorithms with running time $2^{n^{o(1)}}$, for a wide range of problems on $\{K_{t,t}, W_{t\times t}\}$-induced minor free graphs. Our result is a substantial generalization of existing bounds for the tree-independence and tree-width on various graph classes, and a partial resolution of the conjecture of Chudnovsky, E S, and Lokshtanov [Arxiv, 2025] that $\{K_{t,t}, W_{t\times t}\}$-induced minor free graphs have poly-logarithmic tree independence number. The generality comes at the cost of a sub-polynomial, rather than poly-logarithmic upper bound. Our result leads to a complete classification of induced-minor closed classes into ones that have sub-polynomial tree-independence, tree-independence equal to $\tilde{O}(\sqrt{n})$, and linear tree-independence.

Problem

• induced-minor-closed
• tree-independence number
• graph classes
• sub-polynomial
• classification

Innovation

• tree-independence number
• induced minor
• sub-polynomial bound
• graph decomposition
• algorithmic graph theory