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Disjoint paths in tournaments. (English) Zbl 1304.05057
Summary: Given $$k$$ pairs of vertices $$(s_i, t_i)\,(1 \leq i \leq k)$$ of a digraph $$G$$, how can we test whether there exist $$k$$ vertex-disjoint directed paths from $$s_i$$ to $$t_i$$ for $$1 \leq i \leq k$$? This is NP-complete in general digraphs, even for $$k = 2$$ [S. Fortune et al., Theor. Comput. Sci. 10, 111–121 (1980; Zbl 0419.05028)], but for $$k = 2$$ there is a polynomial-time algorithm when $$G$$ is a tournament (or more generally, a semicomplete digraph), due to J. Bang-Jensen and C. Thomassen [SIAM J. Discrete Math. 5, No. 3, 366–376 (1992; Zbl 0759.05041)]. Here we prove that for all fixed $$k$$ there is a polynomial-time algorithm to solve the problem when $$G$$ is semicomplete.

##### MSC:
 05C20 Directed graphs (digraphs), tournaments 05C38 Paths and cycles 05C85 Graph algorithms (graph-theoretic aspects) 90C39 Dynamic programming
##### Keywords:
tournament; routing; dynamic programming; disjoint paths
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##### References:
 [1] Bang-Jensen, J.; Thomassen, C., A polynomial algorithm for the 2-path problem for semicomplete digraphs, SIAM J. Discrete Math., 5, 366-376, (1992) · Zbl 0759.05041 [2] Fortune, S.; Hopcroft, J.; Wyllie, J., The directed subgraphs homeomorphism problem, Theoret. Comput. Sci., 10, 111-121, (1980) · Zbl 0419.05028 [3] Fradkin, A.; Seymour, P., Edge-disjoint paths in digraphs with bounded independence number, J. Combin. Theory Ser. B, 110, 19-46, (2015) · Zbl 1302.05067 [4] Robertson, N.; Seymour, P. D., Graph minors. XIII. the disjoint paths problem, J. Combin. Theory Ser. B, 63, 65-110, (1995) · Zbl 0823.05038
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