The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties, but the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milanič, Munaro, Štorgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, every $\{P_t,K_{\ell,\ell}\}$-free graph has bounded tree-independence number. We prove this conjecture for $t=5$ by showing that every $\{P_5,K_{\ell,\ell}\}$-free graph has tree-independence number at most $4\ell$. We also obtain related bounds for the weaker parameter of $α$-degeneracy.

翻译：图的树独立数定义为在所有树分解中，单个袋子所含独立集的最大尺寸的最小值。具有有界树独立数的图类拥有显著的结构和算法特性，但即使在相当受限的图类中该参数也可能无界。特别地，诱导双团$K_{\ell,\ell}$的存在迫使树独立数至少为$\ell$。这引出一个问题：在自然遗传图类中，大型诱导双团是否成为有界树独立数的唯一障碍？Dallard、Krnc、Kwon、Milanič、Munaro、Štorgel和Wiederrecht提出猜想：对所有正整数$t$和$\ell$，每个$\{P_t,K_{\ell,\ell}\}$-自由图都具有有界树独立数。我们通过证明每个$\{P_5,K_{\ell,\ell}\}$-自由图的树独立数至多为$4\ell$，验证了该猜想在$t=5$情形下的正确性。本文还获得了关于较弱参数$α$-退化度的相关界值。