Disjoint paths in graphs Original Research Article

Suppose that image are pairs of vertices of a graph. When can one choose a path between image and image for each i, all pairwise edge-disjoint? Mengerʹs theorem answers this when image take only two distinct values, but the general problem is unsolved. We settle the two next simplest cases, (i) when image, and (ii) when image take only three distinct values—the solution to this is obtained by applying a theorem of Mader. We obtain both good characterizations and good algorithms for these problems. The analogous “vertex-disjoint” problems are also solved.