A Maximum Dicut in a Digraph Induced by a Minimal Dominating Set


Let \(G = (V,A)\) be a simple directed graph and let \(S\subseteq V \) be a subset of the vertex set \(V \). The set \(S \) is called dominating if for each vertex \(j\in V\setminus S\) there exist at least one \(i\in S \) and an edge from \(i \) to \(j\). A dominating set is called (inclusion) minimal if it contains no smaller dominating set. A dicut \(\{S\rightarrow \overline {S}\} \) is a set of edges \((i,j)\in A \) such that \(i\in S \) and \(j\in V\setminus S \). The weight of a dicut is the total weight of all its edges. The article deals with the problem of finding a dicut \(\{S\rightarrow \overline {S}\} \) with maximum weight among all minimal dominating sets.

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  1. 1

    N. Christofides, Graph Theory: An Algorithmic Approach (Academic Press, London, 1975; Mir, Moscow, 1978).

    Google Scholar 

  2. 2

    R. M. Karp, “Reducibility among Combinatorial Problems,” in Complexity of Computer Computations (Proceedings of Symposium CCC, Yorktown Heights, USA, March 20–22, 1972) (Plenum Press, New York, 1972), pp. 85–103.

  3. 3

    A. Ageev, R. Hassin, and M. Sviridenko, “A \(0.5 \)-Approximation Algorithm for Max Dicut with Given Sizes of Parts,” SIAM J. Discrete Math. 14 (2), 246–255 (2001).

    MathSciNet  Article  Google Scholar 

  4. 4

    J. Lee, V. Nagarajan, and X. Shen, “Max-Cut under Graph Constraints,” in Integer Programming and Combinatorial Optimization: Proceedings of 18th International Conference (Liège, Belgium, June 1–3, 2016) (Springer, Cham, 2016), pp. 50–62 [Lecture Notes Computer Science, Vol. 9682].

  5. 5

    G. A. Cheston, G. Fricke, S. T. Hedetniemi, and D. P. Jacobs, “On the Computational Complexity of Upper Fractional Domination,” Discrete Appl. Math. 27 (3), 195–207 (1990).

    MathSciNet  Article  Google Scholar 

  6. 6

    N. Boria, F. Della Croce, and V. Th. Paschosdef, “On the Max Min Vertex Cover Problem,” Discrete Appl. Math. 196, 62–71 (2015).

    MathSciNet  Article  Google Scholar 

  7. 7

    R. Yu. Simanchev, I. V. Urazova, V. V. Voroshilov, V. V. Karpov, and A. A. Korableva, “Selection of the Key-Indicator System for the Economic Security of a Region by Using a \((0,1) \)-Programming Model,” Vestnik Omsk. Gos. Univ. Ser. Ekonomika 17 (3), 170–179 (2019).

    Google Scholar 

  8. 8

    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness (Freeman, San Francisco, 1979; Mir, Moscow, 1982).

    Google Scholar 

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Correspondence to V. V. Voroshilov.

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Translated by Ya.A. Kopylov

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Voroshilov, V.V. A Maximum Dicut in a Digraph Induced by a Minimal Dominating Set. J. Appl. Ind. Math. 14, 792–801 (2020). https://doi.org/10.1134/S199047892004016X

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  • directed graph
  • weighted graph
  • maximum dicut
  • inclusion minimal dominating set