Processor Efficient Parallel Algorithms for the Two Disjoint Paths Problem and for Finding a Kuratowski Homeomorph

The authors give a parallel algorithm for finding vertex disjoint $s_1 ,t_1 $ and $s_2 ,t_2 $ paths in an undirected graph G. An important step in solving the general problem is solving the planar case. A new structural property yields the parallelization, as well as a simpler linear-time sequential algorithm for this case. The algorithm is extended to the nonplanar case by giving a parallel algorithm for finding a Kuratowski homeomorph, and, in particular, a homeomorph of $K_{3,3} $, in a nonplanar graph. The algorithms are processor efficient; in each case, the processor-time product of the algorithms is within a polylogarithmic factor of the best-known sequential algorithm.