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Efficient Solvability of the Weighted Vertex Coloring Problem for Some Hereditary Class of Graphs with \(\boldsymbol {5}\) -Vertex Prohibitions

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Abstract

We consider the problem of minimizing the number of colors in the colorings of the vertices of a given graph so that, to each vertex there is assigned some set of colors whose number is equal to the given weight of the vertex; and adjacent vertices receive disjoint sets. For all hereditary classes defined by a pair of forbidden induced connected subgraphs on \(5 \) vertices but four cases, the computational complexity of the weighted vertex coloring problem with unit weights is known. We prove the polynomial solvability on the sum of the vertex weights for this problem and the intersection of two of the four open cases. We hope that our result will be helpful in resolving the computational complexity of the weighted vertex coloring problem in the above-mentioned forbidden subgraphs.

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Funding

The authors were supported by the Russian Foundation for Basic Research (project no. 18–31–2001–mol-a-ved).

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Authors and Affiliations

  1. National Research University “Higher School of Economics”, Nizhny Novgorod, 603155, Russia

    D. V. Gribanov, D. S. Malyshev & D. B. Mokeev

  2. Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, 603950, Russia

    D. V. Gribanov, D. S. Malyshev & D. B. Mokeev

Authors
  1. D. V. Gribanov
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  2. D. S. Malyshev
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  3. D. B. Mokeev
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Corresponding authors

Correspondence to D. V. Gribanov, D. S. Malyshev or D. B. Mokeev.

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Translated by Ya.A. Kopylov

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Gribanov, D.V., Malyshev, D.S. & Mokeev, D.B. Efficient Solvability of the Weighted Vertex Coloring Problem for Some Hereditary Class of Graphs with \(\boldsymbol {5}\) -Vertex Prohibitions. J. Appl. Ind. Math. 14, 480–489 (2020). https://doi.org/10.1134/S1990478920030072

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  • DOI: https://doi.org/10.1134/S1990478920030072

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