Computational Complexity of the Chromatic Art Gallery Problem for Orthogonal Polygons The art gallery problem is to find a set of guards who together can observe every point of the interior of a polygon P. We study a chromatic variant of the problem, where each guard is assigned one of k distinct colors. The chromatic art gallery problem is to find a guard set for P such that no two guards with the same color have overlapping visibility regions. We study the decision version of this problem for orthogonal polygons with rvisibility when the number of colors is (k2). Here, two points are rvisible if the smallest axisaligned rectangle containing them lies entirely within the polygon. In this paper, it is shown that determining whether there is an rvisibility guard set for an orthogonal polygon with holes such that no two guards with the same color have overlapping visibility regions is NPhard when the number of colors is (k2).

Contrast in greyscales of graphs In this paper, we present the notion of greyscale of a graph, as a colouring of its vertices that uses colours from the continuous spectrum 0,xc2xa01. Any greyscale induces another colouring by assigning to each edge the nonnegative difference between the colours of its vertices. These edge colours are ordered in lexicographical increasing order and make up the contrast vector of the greyscale. The aim of the contrast problem for the given graph is to find the maximum contrast vector and the greyscales that give rise to it, namely the maximum contrast greyscales. The NPcompleteness of this problem is stated by means of a functional relation between the chromatic number and the first component of the maximum contrast vector, named the lightest tone. Thus, we introduce the notion of lightest tone as a new invariant of the graph. The underlying structure of values of maximum contrast greyscales is addressed and we prove that they are linked by rational number sets, which are algorithmically determined. The restricted maximum contrast problem, that is, greyscales with prefixed extreme tones, is also defined and solved in polynomial time for different families of bipartite graphs.