Processor efficient parallel algorithms for the two disjoint paths problem, and for finding a Kuratowski homeomorph

Given a graph G and two pairs of vertices s₁, t₁ and s₂, t₂, the two disjoint paths problem asks for vertex-disjoint paths connecting s_i with t_i, i=1, 2. A fast parallel (NC) algorithm is given for this problem, which has applications in certain routing situations. If G is nonplanar, an algorithm that finds a Kuratowski homeomorph in G (i.e. a subgraph homeomorphic to K_3.3 or K₅) is given. This complements the known NC planarity algorithms, which give a planar embedding in the positive case; the algorithm provides a certificate of nonplanarity in the negative case. Both algorithms are processor efficient; in each case, the processor-time product is within a polylogarithmic factor of the best known sequential algorithm