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 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 \(k\ge 1\). 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.