Induced disjoint paths in AT-free graphs

Golovach, P.A.; Paulusma, D.; van Leeuwen, E.J.

Abstract

Paths P1, . . . , Pk in a graph G = (V, E) are mutually induced if any two distinct Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (si, ti) contains k mutually induced paths Pi such that each Pi connects si and ti. This is a classical graph problem that is NP-complete even for k = 2. We study it for AT-free graphs. Unlike its subclasses of permutation graphs and cocomparability graphs, the class of AT-free graphs has no known characterisation by a geometric intersection model. However, by a new, structural analysis of the behaviour of Induced Disjoint Paths for AT-free graphs, we prove that it can be solved in polynomial time for AT-free graphs even when k is part of the input. This is in contrast to the situation for other well-known graph classes, such as planar graphs, claw-free graphs, or more recently, (theta,wheel)-free graphs (JCTB 2021), for which such a result only holds if k is fixed. As a consequence of our main result, the problem of deciding if a given ATfree graph contains a fixed graph H as an induced topological minor admits a polynomial-time algorithm. In addition, we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter |VH|, even on a subclass of AT-free graphs, namely cobipartite graphs. We also show that the problems k-in-a-Path and k-in-a-Tree are polynomialtime solvable on AT-free graphs even if k is part of the input. These problems are to test if a graph has an induced path or induced tree, respectively, spanning k given vertices.

Citation

Golovach, P., Paulusma, D., & van Leeuwen, E. (2022). Induced disjoint paths in AT-free graphs. Journal of Computer and System Sciences, 124, 170-191. https://doi.org/10.1016/j.jcss.2021.10.003

• Journal Article Type: Article
• Acceptance Date: Oct 18, 2021
• Online Publication Date: Oct 29, 2021
• Publication Date: 2022-03
• Deposit Date: Oct 23, 2021
• Publicly Available Date: Oct 29, 2022
• Journal: Journal of Computer and System Sciences
• Print ISSN: 0022-0000
• Electronic ISSN: 1090-2724
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 124
• Pages: 170-191
• DOI: https://doi.org/10.1016/j.jcss.2021.10.003
• Public URL: https://durham-repository.worktribe.com/output/1233633