Abstract

This paper considers the two-step colouring problem for an undirected connected graph. The problem is about colouring the graph in a given number of colours so that no pair of vertices at a distance of 1 or 2 between each other has the same colour. Also the corresponding recognition problem is set. The problem is closely related to the classical graph colouring problem. In the article, the polynomial reduction of the problems to one another is analyzed and proved. In particular, this allows us to prove the NP-completeness of the two-step colouring problem. Also we specify some of its properties. The two-step colouring problem as applied to rectangular grid graphs is considered separately. The maximum vertex degree in these graphs ranges from 0 to 4. The function of two-vertex colouring in the minimum possible number of colours was defined and proved for each case. The resulting functions are drawn so that each vertex is coloured independently from others. If the vertices are examined in a sequence, the time for colouring a rectangular grid graph will be polynomial.

中文翻译：

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两步图着色问题的NP完备性和一个多项式子类

摘要

本文考虑了无向连通图的两步着色问题。问题在于要以给定数量的颜色为图形着色，以使彼此之间距离为1或2的一对顶点都不具有相同的颜色。还设置了相应的识别问题。该问题与经典图着色问题密切相关。本文对问题的多项式约简进行了分析和证明。特别是，这使我们能够证明两步着色问题的NP完全性。我们还指定了它的一些属性。分别考虑应用于矩形网格图的两步着色问题。这些图中的最大顶点度为0到4。定义并证明了每种情况下最小数量的双顶点着色功能。绘制结果函数，以便每个顶点彼此独立着色。如果按顺序检查顶点，则为矩形网格图着色的时间将是多项式。