Induced Disjoint Paths in Circular-Arc Graphs in Linear Time

The Induced Disjoint Paths problem is to test whether a graph G with k distinct pairs of vertices (s_i,t_i) contains paths P_1,…,P_k such that P_i connects s_i and t_i for i=1,…,k, and P_i and P_j have neither common vertices nor adjacent vertices (except perhaps their ends) for 1<=i < j<=k. We present a linear-time algorithm for Induced Disjoint Paths on circular-arc graphs. For interval graphs, we exhibit a linear-time algorithm for the generalization of Induced Disjoint Paths where the pairs (s_i,t_i) are not necessarily distinct.