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Degree constrained 2-partitions of semicomplete digraphs. (English) Zbl 1401.05129
Summary: A 2-partition of a digraph $$D$$ is a partition $$(V_1, V_2)$$ of $$V(D)$$ into two disjoint non-empty sets $$V_1$$ and $$V_2$$ such that $$V_1 \cup V_2 = V(D)$$. A semicomplete digraph is a digraph with no pair of non-adjacent vertices. We consider the complexity of deciding whether a given semicomplete digraph has a 2-partition such that each part of the partition induces a (semicomplete) digraph with some specified property. J. Bang-Jensen and F. Havet [Theor. Comput. Sci. 636, 85–94 (2016; Zbl 1342.68150)] and J. Bang-Jensen et al. [ibid. 640, 1–19 (2016; Zbl 1345.68168)] determined the complexity of 120 such 2-partition problems for general digraphs. Several of these problems are NP-complete for general digraphs and thus it is natural to ask whether this is still the case for well-structured classes of digraphs, such as semicomplete digraphs. This is the main topic of the paper. More specifically, we consider 2-partition problems where the set of properties are minimum out-, minimum in- or minimum semi-degree. Among other results we prove the following:
i) For all integers $$k_1, k_2 \geq 1$$ and $$k_1 + k_2 \geq 3$$ it is NP-complete to decide whether a given digraph $$D$$ has a 2-partition $$(V_1, V_2)$$ such that $$D \langle V_i \rangle$$ has out-degree at least $$k_i$$ for $$i = 1, 2$$.
ii) For every fixed choice of integers $$\alpha, k_1, k_2 \geq 1$$ there exists a polynomial algorithm for deciding whether a given digraph of independence number at most $$\alpha$$ has a 2-partition $$(V_1, V_2)$$ such that $$D \langle V_i \rangle$$ has out-degree at least $$k_i$$ for $$i = 1, 2$$.
iii) For every fixed integer $$k \geq 1$$ there exists a polynomial algorithm for deciding whether a given semicomplete digraph has a 2-partition $$(V_1, V_2)$$ such that $$D \langle V_1 \rangle$$ has out-degree at least one and $$D \langle V_2 \rangle$$ has in-degree at least $$k$$.
iv) It is NP-complete to decide whether a given semicomplete digraph $$D$$ has a 2-partition $$(V_1, V_2)$$ such that $$D \langle V_i \rangle$$ is a strong tournament.

MSC:
 05C20 Directed graphs (digraphs), tournaments 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 68Q25 Analysis of algorithms and problem complexity
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References:
 [1] Alon, N., Disjoint directed cycles, J. Combin. Theory Ser. B, 68, 167-178, (1996) · Zbl 0861.05037 [2] N. Alon, J. Bang-Jensen, S. Bessy, Out-colourings of digraphs, submitted for publication, 2017. [3] Bang-Jensen, J.; Gutin, G., Digraphs: theory, algorithms and applications, (2009), Springer Verlag London · Zbl 1170.05002 [4] Bang-Jensen, J.; Havet, F., Finding good 2-partitions of digraphs I. hereditary properties, Theoret. Comput. Sci., 636, 85-94, (2016) · Zbl 1342.68150 [5] Bang-Jensen, J.; Cohen, N.; Havet, F., Finding good 2-partitions of digraphs II. enumerable properties, Theoret. Comput. Sci., 640, 1-19, (2016) · Zbl 1345.68168 [6] Bang-Jensen, J.; Nielsen, M. H., Finding complementary cycles in locally semicomplete digraphs, Discrete Appl. Math., 146, 245-256, (2005) · Zbl 1055.05086 [7] Guo, Y.; Volkmann, L., On complementary cycles in locally semicomplete digraphs, Discrete Math., 135, 121-127, (1994) · Zbl 0816.05036 [8] Kühn, D.; Osthus, D.; Townsend, T., Proof of a tournament partition conjecture and an application to 1-factors with prescribed cycle length, Combinatorica, 36, 451-469, (2016) · Zbl 1389.05058 [9] Li, H.; Shu, J., The partition of a strong tournament, Discrete Math., 290, 211-220, (2005) · Zbl 1069.05037 [10] Lichiardopol, N., Vertex-disjoint subtournaments of prescribed minimum outdegree or minimum semi-degree: proof for tournaments of a conjecture of stiebitz, Int. J. Comb., 1-9, (2012) · Zbl 1236.05095 [11] McCuaig, W., Intercyclic digraphs, (Graph Structure Theory, Seattle, WA, 1991, Contemp. Math., vol. 147, (1993), American Mathematical Society), 203-245 · Zbl 0789.05042 [12] Moon, J. W., Topics on tournaments, (1968), Holt, Rinehart and Winston New York · Zbl 0191.22701 [13] Reid, K. B., Two complementary circuits in two-connected tournaments, (Cycles in Graphs, North-Holland Mathematical Studies, vol. 115, (1985), North-Holland Amsterdam), 321-334 · Zbl 0573.05031 [14] Stiebitz, M., Decomposition of graphs and digraphs, (KAM Series in Discrete Mathematics-Combinatorics-Operations Research-Optimization, vol. 95-309, (1995)), 56-59 [15] Thomassen, C., Disjoint cycles in digraphs, Combinatorica, 3, 393-396, (1983) · Zbl 0527.05036 [16] Schaefer, T. J., The complexity of satisfiability problems, (Proceedings of the 10th Annual ACM Symposium on Theory of Computing, (1978), ACM New York), 216-226 · Zbl 1282.68143
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