We continue the study of graph classes in which the treewidth can only be large due to the presence of a large clique, and, more specifically, of graph classes with bounded tree-independence number. In [Dallard, Milani\v{c}, and \v{S}torgel, Treewidth versus clique number. {II}. Tree-independence number, 2022], it was shown that the Maximum Weight Independent Packing problem, which is a common generalization of the Independent Set and Induced Matching problems, can be solved in polynomial time provided that the input graph is given along with a tree decomposition with bounded independence number. We provide further examples of algorithmic problems that can be solved in polynomial time under this assumption. This includes, for all even positive integers $d$, the problem of packing subgraphs at distance at least $d$ (generalizing the Maximum Weight Independent Packing problem) and the problem of finding a large induced sparse subgraph satisfying an arbitrary but fixed property expressible in counting monadic second-order logic. As part of our approach, we generalize some classical results on powers of chordal graphs to the context of general graphs and their tree-independence numbers.
翻译:我们继续研究图形类,因为树枝之所以大,是因为存在一个大区块,更具体地说,是树状图类中树状独立的数字。在[Dallard, Milani\v{c}和\v{S}torgel, Treewidth对clique number. {II},树状独立号2022)中,树状独立号显示,最大重量独立包装问题是独立集和诱导的匹配问题的一个常见的概略,在多元时间可以解决,条件是输入图与带条独立号的树状分解状态同时提供。我们在此假设下提供了在多元时间可以解决的算法问题的更多例子。这包括,对于所有正整数的整数美元来说,至少以美元(将最大重量独立集成的问题)包装子体的问题,以及找到一个大型导引的稀释子体子体的问题,在计算某种任意但固定的属性时,可以明确表示独立号的独立号独立号。我们从一般的直径直径图上将其直径直径直径直到直径直径直径直径直径直径直径直径直径直径直图的逻辑。