Efficient solvability of the weighted vertex coloring problem for some hereditary class of~graphs with $5$-vertex prohibitions
Diskretnyj analiz i issledovanie operacij, Tome 27 (2020) no. 3, pp. 71-87.

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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. Illustr. 1, bibliogr. 18.
Keywords: weighted vertex coloring problem, hereditary class, computational complexity.
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D. V. Gribanov; D. S. Malyshev; D. B. Mokeev. Efficient solvability of the weighted vertex coloring problem for some hereditary class of~graphs with $5$-vertex prohibitions. Diskretnyj analiz i issledovanie operacij, Tome 27 (2020) no. 3, pp. 71-87. http://geodesic.mathdoc.fr/item/DA_2020_27_3_a3/

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