Contrast in Greyscales of Graphs

TL;DR

This paper introduces the concept of maximum contrast greyscales in graphs, explores their relation to graph chromatic number, proves the problem's NP-completeness, and discusses methods for computing maximum contrast vectors.

Contribution

It defines maximum contrast greyscales, establishes their connection to graph chromatic number, proves the problem is NP-complete, and surveys methods for computing these vectors.

Findings

Maximum contrast vectors are related to graph chromatic number.

The maximum contrast problem is NP-complete.

Finite bounds exist for the values of maximum contrast greyscales.

Abstract

A greyscale $f$ of a graph $G(V,E)$ is a mapping from $V$ to the interval $[0,1]$ such that $\{0, 1\} \subseteq Im(f)$. This function $f$ induces another mapping $\widehat{f}$ on $E$ by assigning to each edge the non-negative difference of the values of $f$ on its vertices. The contrast vector $cont(G,f)$ is defined as the vector $(\widehat{f}(e_1), \widehat{f}(e_2), \ldots, \widehat{f}(e_m))$ for all edges $e_i$ of $G$ in such a way that $\widehat{f}(e_i) \leq \widehat{f}(e_{i+1})$ for $i= 1, 2, \dots, m-1$. The concept of maximum contrast vector is presented by using the lexicographical ordering in the set of contrast vectors of all possible greyscales defined on $G$ and a greyscale that gives rise to a maximum contrast vector is named maximum contrast greyscale. The relation between finding the maximum contrast vector for the graph $G$ and the chromatic number of $G$ is established. Thus the maximum contrast problem is an NP-complete problem. However, the set of values of any maximum contrast greyscale for any graph is bounded by a finite set which is given. Several methods to compute the maximum contrast vector with some restrictions in are collected in this paper.