Suppose that (s1, t1),…,(sk, tk) are pairs of vertices of a graph. When can one choose a path between s1 and t1 for each i, all pairwise edge-disjoint? Menger's theorem answers this when s1,…,sk, t1,…,tk take only two distinct values, but the general problem is unsolved. We settle the two next simplest cases.
1.
(i) when k = 2, and
2.
(ii) when s1,…,sk, t1,…,tk 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.