Induced Disjoint Paths in Circular-Arc Graphs in Linear Time

The Induced Disjoint Paths problem is to test whether an graph G on n vertices with k distinct pairs of vertices (si, ti) contains paths P1, . . . , Pk such that Pi connects si and ti for i = 1, . . . , k, and Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their ends) for 1 ≤ i < j ≤ k. We present a linear-time algorithm that solves Induced Disjoint Paths and finds the corresponding paths (if they exist) on circular-arc graphs. For interval graphs, we exhibit a linear-time algorithm for the generalization of Induced Disjoint Paths where the pairs (si, ti) are not necessarily distinct. In both cases, if a representation of the graph is given, then the algorithms run in O(n+k) time.