On three parameters of invisibility graphs

Abstract

The invisibility graph I(X) of a set X ⊆ Rd is a (possibly infinite) graph whose vertices are the points of X and two vertices are connected by an edge if and only if the straight-line segment connecting the two corresponding points is not fully contained in X . We consider the following three parameters of a set X : the clique number ω(I(X)), the chromatic number χ(I(X)) and the inimum number γ(X) of convex subsets of X that cover X. We settle a conjecture of Matousek and Valtr claiming that for every planar set X, γ(X) can be bounded in terms of χ(I(X)). As a part of the proof we show that a disc with n one-point holes near its boundary has χ(I(X)) ≥ log log(n) but ω(I(X)) = 3. We also find sets X in R5 with χ(I(X)) = 2, but γ(X) arbitrarily large.