Efficient Solvability of the Weighted Vertex Coloring Problem for Some Hereditary Class of Graphs with 5-Vertex Prohibitions

Abstract: We consider the problem of minimizing the number of colors in the colorings of thevertices of a given graph so that, to each vertex there is assigned some set of colors whose numberis equal to the given weight of the vertex; and adjacent vertices receive disjoint sets. For allhereditary classes defined by a pair of forbidden induced connected subgraphs on 5vertices but four cases, the computationalcomplexity of the weighted vertex coloring problem with unit weights is known. We prove thepolynomial solvability on the sum of the vertex weights for this problem and the intersection of twoof the four open cases. We hope that our result will be helpful in resolving the computationalcomplexity of the weighted vertex coloring problem in the above-mentioned forbidden subgraphs. © 2020, Pleiades Publishing, Ltd.