Disjoint Paths—A Survey

We describe without proof polynomially bounded algorithms for the following problems: (i) (k is a fixed integer, andS is a fixed surface). With input a graph G which may be drawn in S and k pairs of vertices of G, decide if there are k vertex-disjoint paths of G, each joining one of the pairs of vertices; (ii) (His a fixed planar graph). With input a graph G, decide if G can be reduced to a graph isomorphic to H by deletion and contraction of edges.