Induced Disjoint Paths and Connected Subgraphs for H-Free Graphs

Abstract

Paths $$P^1,\ldots,P^k$$ 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.