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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:
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