Publication details

ON THREE MEASURES OF NON-CONVEXITY

Authors

CIBULKA Josef KORBELÁŘ Miroslav KYNCL Jan MESZAROS Viola STOLAR Rudolf CIBULKA J. VALTR Pavel

Year of publication 2017
Type Article in Periodical
Magazine / Source Israel journal of mathematics
MU Faculty or unit

Faculty of Science

Citation
Web Full Text
Doi http://dx.doi.org/10.1007/s11856-017-1467-1
Keywords CONVEX-SETS; DECOMPOSITION THEOREMS; SUBSETS; UNION; PLANE
Description The invisibility graph I (X) of a set X subset of 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 omega(I(X)), the chromatic number chi(I(X)) and the convexity number gamma(X), which is the minimum number of convex subsets of X that cover X. We settle a conjecture of Matousek and Valtr claiming that for every planar set X, gamma(X) can be bounded in terms of chi(I( X)). As a part of the proof we show that a disc with n one-point holes near its boundary has chi(I(X)) >= log log(n) but omega (I(X)) = 3. We also find sets X in R-5 with chi(X) = 2, but gamma( X) arbitrarily large.
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