Induced Disjoint Paths and Connected Subgraphs for H-Free Graphs

Paths Pj, . . . , P-k in a graph G = (V, E) are mutually induced if any two distinct P-i and P 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.