Disjoint paths and connected subgraphs for H-free graphs

The well-known DISJOINT PATHS problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct vertex pairs. We determine, with an exception of two cases, the complexity of the DISJOINT PATHS problem for H-free graphs. If k is fixed, we obtain the k-DISJOINT PATHS problem, which is known to be polynomial-time solvable on the class of all graphs for every k≥1. The latter does no longer hold if we need to connect vertices from terminal sets instead of terminal pairs. We completely classify the complexity of k-DISJOINT CONNECTED SUBGRAPHS for H-free graphs, and give the same almost-complete classification for DISJOINT CONNECTED SUBGRAPHS for H-free graphs as for DISJOINT PATHS. Moreover, we give exact algorithms for DISJOINT PATHS and DISJOINT CONNECTED SUBGRAPHS on graphs with n vertices and m edges that have running times of O(2ⁿn²k) and O(3ⁿkm), respectively.