Polynomial-time algorithms for the Longest Induced Path and Induced Disjoint Paths problems on graphs of bounded mim-width

TL;DR

This paper presents the first polynomial-time algorithms for certain complex graph problems on graphs with bounded mim-width, expanding efficient solutions to various graph classes.

Contribution

It introduces $n^{O(w)}$-time algorithms for the Longest Induced Path and related problems on graphs with bounded mim-width, when a decomposition is provided.

Findings

Polynomial-time algorithms for Longest Induced Path and Induced Disjoint Paths.

Applicable to multiple graph classes including interval, circular arc, and permutation graphs.

Results extend tractability to problems not locally checkable.

Abstract

We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give $n^{\mathcal{O}(w)}$-time algorithms on graphs of mim-width at most $w$, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and $H$-Induced Topological Minor for fixed $H$. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Permutation and Circular Permutation graphs, Convex graphs, $k$-Trapezoid, Circular $k$-Trapezoid, $k$-Polygon, Dilworth-$k$ and Co-$k$-Degenerate graphs for fixed $k$.