Tree decompositions with bounded independence number: beyond independent sets

TL;DR

This paper explores graph classes with bounded tree-independence number, demonstrating polynomial-time solutions for various packing problems and subgraph detection, extending classical results to broader graph categories.

Contribution

It introduces new polynomial-time algorithms for packing and subgraph problems in graphs with bounded tree-independence number, generalizing previous results on chordal graphs.

Findings

Polynomial-time algorithms for packing subgraphs at fixed distances.

Efficient detection of large induced subgraphs with fixed MSO properties.

Extension of classical chordal graph results to general graphs.

Abstract

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.