Two-Trees Optimal T-Join and Integral Packing of T-Cuts

Let G be an undirected graph, T an even subset of vertices and F an optimal T-join, which is a forest of two trees. The main theorem of this paper characterizes the cases, where (G, T) has an optimal packing of T-cuts which is integral. This theorem unifies and generalizes a theorem of P. Seymour on packing of T-cuts and a theorem of A. Frank on planar edge disjoint paths. It also solves positively a conjecture by A, Frank. The proof of the main theorem implies a polynomial algorithm for optimal integral packing of T-cuts for the case where the optimal T-join consists of two trees. This algorithm is in fact a simple post-optimality method that can be applied to existing algorithms for 12 integral packing of T-cuts and also solves polynomially a certain planar integral multicommodity now problem.