Abstract
Suppose that ( s 1, t 1),…,( s k , t k ) are pairs of vertices of a graph. When can one choose a path between s 1 and t 1 for each i, all pairwise edge-disjoint? Menger's theorem answers this when s 1,…, s k , t 1,…, t k 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 s 1,…, s k , t 1,…, t k 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.
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