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.

中文翻译：

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具有$$ \ boldsymbol {5} $$-顶点禁止的某些遗传类图的加权顶点着色问题的有效可解性

摘要

我们考虑使给定图的顶点着色中的颜色数量最小化的问题，以便为每个顶点分配一些颜色，其数量等于给定顶点权重；并且相邻的顶点接收不相交的集合。对于由\（5 \）顶点上的一对禁止诱导子图定义的所有遗传类， 但有四种情况，已知单位权重的加权顶点着色问题的计算复杂性。我们证明了该问题的顶点权重之和与四个开放情况中的两个相交的多项式的可解性。我们希望我们的结果将有助于解决上述禁止子图中加权顶点着色问题的计算复杂性。