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.

翻译：暂无翻译