发表机构
Department of Mathematical, Physical and Computer Sciences, University of Parma（帕尔马大学数学、物理与计算机科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文证明遗传图类的强次线性树独立数与团数依赖的强次线性树宽等性质等价，并给出计算强次线性独立数树分解的亚指数时间算法。
AI 中文摘要
我们建立了Chudnovsky、E S和Lokshtanov（arXiv 2025）关于树宽和树独立数的一个近期结果的强次线性对应版本。具体而言，我们证明了一个遗传图类具有强次线性树独立数，当且仅当对于每个固定的团数界，其有界团数的图具有强次线性树宽。事实上，这是一个更广泛的等价定理的一部分。对于遗传类，这些条件还等价于具有团数依赖的多项式膨胀，等价于承认大小由$K\omega(G)^s |V(G)|^{1-\beta}$界定的平衡分离器（其中$K,s,\beta>0$为固定常数），以及等价于承认强次线性大小（等价地，权重）的基于团的平衡分离器。因此，我们表明所有这些性质——它们分别源于亚指数时间精确算法和多项式时间近似方案的研究——实际上描述了相同的遗传图类。作为我们等价定理的一个推论，我们还证明了每个具有强次线性树独立数的遗传类$\mathcal C$都承认一个亚指数时间算法，该算法给定$G\in\mathcal C$，计算$G$的一个具有强次线性独立数的树分解。
英文摘要
We establish a strongly sublinear counterpart of a recent result of Chudnovsky, E S, and Lokshtanov (arXiv 2025) on treewidth and tree-independence number. Namely, we prove that a hereditary graph class has strongly sublinear tree-independence number if and only if, for every fixed clique bound, its graphs of bounded clique number have strongly sublinear treewidth. In fact, this is part of a broader equivalence theorem. For hereditary classes, these conditions are also equivalent to having clique-dependent polynomial expansion, to admitting balanced separators whose size is bounded by $Kω(G)^s |V(G)|^{1-β}$ for fixed $K,s,β>0$, and to admitting balanced clique-based separators of strongly sublinear size (equivalently, weight). Thus, we show that all these properties, which arose independently in the study of subexponential-time exact algorithms and polynomial-time approximation schemes, in fact describe the same hereditary graph classes. As a consequence of our equivalence theorem, we also show that every hereditary class $\mathcal C$ with strongly sublinear tree-independence number admits a subexponential-time algorithm that, given $G\in\mathcal C$, computes a tree decomposition of $G$ with strongly sublinear independence number.
Comments29 pages
