On three measures of non-convexity
release_ypmp26tcnraqhefximqzusa56u
by
Josef Cibulka, Miroslav Korbelář, Jan Kynčl, Viola
Mészáros, Rudolf Stolař, Pavel Valtr
2015
Abstract
The invisibility graph I(X) of a set X ⊆R^d 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 convexity number γ(X), which is
the minimum number of convex subsets of X that cover X. We settle a
conjecture of Matoušek 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)) >(n) but ω(I(X))=3. We also find sets X in R^5 with
χ(X)=2, but γ(X) arbitrarily large.
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