Mim-Width I. Induced path problems

Abstract

We initialize a series of papers deepening the understanding of algorithmic properties of the width parameter maximum induced matching width (mim-width) of graphs. In this first volume we provide the first polynomial-time algorithms on graphs of bounded mim-width for problems that are not locally checkable. In particular, we give n(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. We contrast these positive results to the fact that problems about finding long non-induced paths remain hard on graphs of bounded mimwidth: We show that HAMILTONIAN CYCLE (and hence HAMILTONIAN PATH) is NP-hard on graphs of linear mim-width 1; this further hints at the expressive power of the mim-width parameter.