诱导子图封闭类具有线性、平方根或次多项式树独立性
Induced-Minor-Closed Classes have Linear, Square-Root, or Sub-Polynomial Tree-Independence
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中文总结 AI 辅助
研究图的诱导子图封闭类的树独立性，证明对每个\(t\in\mathbb{N}\)，图\(G\)要么含特定诱导子图，要么树独立性至多为\(O(2^{O((\log n)^{1-\epsilon})})\)，给出相关算法及分类，推广了现有界并部分解决猜想。
中文摘要 AI 辅助
图\(G\)中的独立集是一组两两不相邻的顶点。图\(G\)的树分解是一对\((T, \chi)\)，其中\(T\)是一棵树，\(\chi: V(T) \rightarrow 2^{V(G)}\)是一个满足两个公理的函数。树分解中\(\chi(x)\)的集合称为树分解的袋。图\(G\)的树独立数是在\(G\)的所有树分解中，由分解的袋所诱导的图的独立集的最大大小的最小值。我们证明对于每个\(t\in\mathbb{N}\)，存在\(\epsilon > 0\)，使得每个图\(G\)要么包含完全二分图\(K_{t,t}\)或墙\(W_{t\times t}\)作为诱导子图，要么树独立性至多为\(O(2^{O((\log n)^{1-\epsilon})})\)。这导致了对于\(\{K_{t,t}, W_{t\times t}\}\)诱导子图自由图上的广泛问题，运行时间为\(2^{n^{o(1)}}\)。我们的结果是对各种图类上树独立性和树宽的现有界的实质性推广，部分解决了Chudnovsky、E S和Lokshtanov的猜想。我们的结果导致了诱导子图封闭类的完整分类，分为具有次多项式树独立性、树独立性等于\(\tilde{O}(\sqrt{n})\)和线性树独立性的类。
英文摘要
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.
