Induced Disjoint Paths and Connected Subgraphs for H-Free Graphs
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Paths P_1,…, P_k in a graph G=(V,E) are mutually induced if any two distinct P_i and P_j have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph G with k pairs of specified vertices (s_i,t_i) contains k mutually induced paths P_i such that each P_i starts from s_i and ends at t_i. This is a classical graph problem that is NP-complete even for k=2. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give almost-complete dichotomies of the computational complexity of both problems for H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. Finally, we give a complete classification of the complexity of the second problem if the number k of terminal sets is fixed, that is, not part of the input.
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