Induced packing treewidth

2026-07-08

AI Summary

This work unifies graph classes defined by forbidden induced subgraphs or induced minors with those characterized by specific tree-decomposition structures by introducing a novel parameter, induced-$\mathcal{H}$-packing treewidth. This parameter measures, for each bag of a tree decomposition, the maximum number of pairwise non-adjacent induced copies of graphs from a family $\mathcal{H}$. It generalizes existing notions such as tree independence number and induced matching treewidth. For various graph families $\mathcal{H}$—including $\{P_3\}$ and all cycles—the paper demonstrates that the Maximum Weight Independent Set (MWIS) problem is solvable in polynomial time on graphs of bounded induced-$\mathcal{H}$-packing treewidth, thereby partially resolving and significantly extending an open question posed by Bodlaender et al. regarding the tractability of MWIS.

Abstract

In this paper, we introduce a framework that aims to unify classes defined by forbidden induced subgraphs or induced minors with classes defined by the existence of certain structured tree decompositions. Let $\mathcal{H}$ be a fixed family of graphs. We define \emph{induced-$\mathcal{H}$-packing treewidth}, a tree-decomposition-based graph parameter that, for each bag, measures the maximum number of pairwise anticomplete induced copies of graphs from $\mathcal{H}$ intersecting that bag. This notion generalizes some previously studied parameters: when $\mathcal{H}=\{P_1\}$, it is equivalent to tree-independence number, and when $\mathcal{H}=\{P_2\}$, it is equivalent to induced matching treewidth. We show that bounded induced-$\mathcal{H}$-packing treewidth yields new algorithmic consequences for a range of choices of $\mathcal{H}$. In particular, we prove the following results for graphs of bounded induced-$\mathcal{H}$-packing treewidth. Our results partially answer and substantially extend a question of Bodlaender, Fomin, and Korhonen [SODA~2026] on the tractability of \textsc{MWIS} for graphs of bounded induced-$\mathcal{H}$-packing treewidth for $\mathcal{H}=\{P_3\}$ and for $\mathcal{H}$ equal to the family of all cycles.

Problem

• induced packing treewidth
• tree decomposition
• forbidden induced subgraphs
• maximum weight independent set
• algorithmic tractability

Innovation

• induced packing treewidth
• tree decomposition
• forbidden induced subgraphs
• maximum weight independent set
• algorithmic graph theory