Given a graph G and pairs \((s_ 1,t_ 1),...,(s_ k,t_ k)\) of vertices, consider the problem of deciding the existence of pairwise vertex-disjoint paths \(P_ 1,...,P_ k\) of G such that \(P_ i\) has ends \(s_ i\) and \(t_ i\). A polynomial algorithm has been shown to exist for \(k=2\) by P. D. Seymour [Discrete Math. 29, 293-309 (1980; Zbl 0457.05043)] and, independently, Y. Shiloach [J. Assoc. Comput. Mach. 27, 445-456 (1980; Zbl 0475.68042)] but whether such exists for \(k\geq 3\) is not known. Here a polynomial algorithm that suffices for planar G and fixed k is described. The second problem considered is that of determining whether a prescribed graph H can be obtained from some subgraph of a graph G by contraction. As before, a polynomial algorithm exists if H is planar. The authors point out that in both cases the algorithms may not be practical because of the large degrees of the polynomials. Detailed proofs of these results are to appear elsewhere.