Abstract
Given a graph G and positive integers b and w, the black-and-white coloring problem asks about the existence of a partial vertex-coloring of G, with b vertices colored black and w white, such that there is no edge between a black and a white vertex. We suggest an improved algorithm for solving this problem on trees.
Highlights
The Black-and-White Coloring (BWC) problem is defined as follows
Given an undirected graph G and positive integers b, w, determine whether there exists a partial vertex-coloring of G such that b vertices are colored black and w vertices in white, such that no black vertex and white vertex are adjacent
We sometimes refer to the optimization version of this problem, in which we are given a graph G and a positive integer b, and have to color b of the vertices in black, so that there will remain as many vertices as possible which are nonadjacent to any of the b vertices
Summary
One application of the BWC problem is to the problem of storing chemical products, where certain pairs of places cannot contain different products. We sometimes refer to the optimization version of this problem, in which we are given a graph G and a positive integer b, and have to color b of the vertices in black, so that there will remain as many vertices as possible which are nonadjacent to any of the b vertices. These latter vertices are to be colored white, and the resulting coloring is optimal. When referring to a BWC, it suffices to refer to its black vertices only
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