Disjoint Paths in Circular Arc Graphs

Arikati, Pandu Rangan, and Manacher (BIT, 31 (1991) 182-193) developed an O(m+n) algorithm to find two vertex disjoint paths in a circular arc graph on n vertices and m edges. We provide an improved solution to this problem: the algorithm presented here is both faster (O(n) time complexity) and simpler than the previous algorithm. The method involves reductions to interval graphs. In an interval graph, the critical notions are unordered paths (vertex disjoint paths from s1, s2 to t1, t2 in either order) and interchangeable paths (existence of both pairs of vertex disjoint paths). We also prove a theorem (which is best possible, in a sense), that guarantees existence of vertex disjoint paths, if arcs are sufficiently dense. Finally, we show that the more general problem of determining the existence of k vertex disjoint paths (from s1,.., sk to t1,..., tk) is NP-complete, where k is part of the input, even restricted co the class of interval graphs.