Induced Disjoint Paths and Connected Subgraphs for H -Free Graphs

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.