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)(si,ti) contains paths P1,…,PkP1,…,Pk such that PiPi connects sisi and titi for i=1,…,ki=1,…,k, and PiPi and PjPj have neither common vertices nor adjacent vertices (except perhaps their ends) for 1≤i