Induced Disjoint Paths in AT-Free Graphs
Paths P 1,…,P k in a graph G = (V,E) are said to be mutually induced if for any 1 ≤ i < j ≤ k, P i and P j have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to test whether a graph G with k pairs of specified vertices (s i ,t i ) contains k mutually induced paths P i such that P i connects s i and t i for i = 1,…,k. This problem is known to be NP-complete already for k = 2. We prove that it can be solved in polynomial time for AT-free graphs even when k is part of the input. As a consequence, the problem of testing whether a given AT-free graph contains some graph H as an induced topological minor admits a polynomial-time algorithm, as long as H is fixed; we show that such an algorithm is essentially optimal by proving that the problem is W-hard, even on a subclass of AT-free graphs, namely cobipartite graphs, when parameterized by |V H |. We also show that the problems k -in-a-Path and k -in-a-Tree can be solved in polynomial time, even when k is part of the input. These problems are to test whether a graph contains an induced path or induced tree, respectively, spanning k given vertices.
KeywordsPolynomial Time Planar Graph Interval Graph Disjoint Path Chordal Graph
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- 1.Belmonte, R., Golovach, P.A., Heggernes, P.: van’t Hof, P., Kaminski, M., Paulusma, D.: Detecting patterns in chordal graphs (preprint)Google Scholar
- 9.Fellows, M.R.: The Robertson-Seymour theorems: A survey of applications. In: Richter, R.B. (ed.) Proc. AMS-IMS-SIAM Joint Summer Research Conference. Contemporary Mathematics, vol. 89, pp. 1–18. Amer. Math. Soc., Providence (1989)Google Scholar
- 11.Golovach, P.A., Paulusma, D., van Leeuwen, E.J.: Induced Disjoint Paths in Claw-Free Graphs. arXiv:1202.4419v1 [cs.DM] (2012)Google Scholar
- 12.Grohe, M., Kawarabayashi, K., Marx, D., Wollan, P.: Finding topological subgraphs is fixed-parameter tractable. In: Proc. STOC, pp. 479–488 (2011)Google Scholar
- 18.Lekkerkerker, C.G., Boland, J.C.: Representation of a finite graph by a set of intervals on the real line. Fund. Math. 51, 45–64Google Scholar