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Edge-disjoint in- and out-branchings in tournaments and related path problems. (English) Zbl 0659.05052
We show that the following problems, are all polynomially solvable for tournaments:
(1) Given a tournament T and \(u,v\in V(T)\), when do there exist edge- disjoint branchings \(F^+_ u\), \(F^-_ u\), such that \(F^+_ u\) is an out-branching rooted at u and \(F^-_ v\) is an in-branching rooted at v? An out-branching rooted at u is a spanning tree which is directed in such a way that each \(x\neq v\) has exactly one edge comming in. An in- branching is defined analogously.
(2) Given a strong tournament T and distinct vertices \(x_ 1\), \(x_ 2\), \(y_ 1\), \(y_ 2\) of V(T), when do there exist edge-disjoint paths \(P_ 1\), \(P_ 2\) connecting \(x_ 1\) to \(y_ 1\) and \(x_ 2\) to \(y_ 2?\)
(3) Given a strong tournament T and distinct vertices a, b, c of V(T), when do there exist edge disjoint paths P, Q, such that P connects a to b and Q connects b to c?
It is well-known that (2) and (3) are NP-complete for general digraphs and we give a proof by C. Thomassen that (1) is also NP-complete for general digraphs. Perhaps a little surprisingly it turns out that problems (1) and (2) are far from trivial, even in the case of tournaments.
Reviewer: J.Bang-Jensen

MSC:
05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
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References:
[1] Bang-Jensen, J, A note on a special case of the 2-path problem for semicomplete digraphs, () · Zbl 0841.05053
[2] Bang-Jensen, J, On the 2-linkage problem for semicomplete digraphs, Ann. discrete math., 41, 23-38, (1989) · Zbl 0664.05034
[3] \scJ. Bang-Jensen and C. Thomassen, A polynomial algorithm for the 2-path problem for semicomplete digraphs, submitted for publication. · Zbl 0759.05041
[4] Edmonds, J, Edge-disjoint branchings, (), 91-96
[5] Edmonds, J, Paths, trees and flowers, Canad. J. math., 17, 449-467, (1965) · Zbl 0132.20903
[6] Fortune, S; Hopcroft, J; Wyllie, J, The directed subgraph homeomorphism problem, Theoret. cimput. sci., 10, 111-121, (1980) · Zbl 0419.05028
[7] Lovász, L, On two minimax theorems in graphs, J. combin. theory ser. B, 21, 96-103, (1976) · Zbl 0337.05115
[8] Chung-fan, Ma; Mao-cheng, Cai, The maximum number of arc-disjoint arborescences in a tournament, J. graph theory, 6, 295-302, (1982) · Zbl 0458.05037
[9] \scC. Thomassen, Private communication.
[10] Thomassen, C, Configurations in graphs of large minimum degree, connectivity or chromatic number, (), 402-412
[11] Thomassen, C, Hamiltonian-connected tournaments, J. combin. theory ser. B, 28, 142-163, (1980) · Zbl 0435.05026
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