On the sharpness of some results relating cuts and crossing numbers

Laurent Beaudou (Author), Drago Bokal (Author)

Abstract

Znano je, da je za majhne povezavne prereze prekrižno število grafa večje ali enako vsoti prekrižnih števil nekoliko dopolnjenih komponent, ki nastanejo ob prerezu. Ob močnejših predpostavkah povezanosti vsake od komponent, ki je bilo formalizirano kot grafovska operacija 'šiv', pa lahko podoben rezultat pokažemo za povezavne prereze poljubne velikosti. Zastavi se naravno vprašanje, ali je ta pogoj potreben. V tem prispevku pokažemo, da šibkejše zahteve za povezanost komponent ne zadoščajo, če prerez vsebuje vsaj štiri povezave. Razlika med vsoto prekrižnih števil komponent in skupnega grafa lahko narašča kvadratično z velikostjo prereza.

Keywords

matematika;teorija grafov;prekrižno število;šiv grafov;prerez v grafih;mathematics;graph theory;crossing number;zip product;graph cuts;

Data

Language:	English
Year of publishing:	2010
Typology:	1.01 - Original Scientific Article
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
UDC:	519.17
COBISS:	15638361
ISSN:	1077-8926
Views:	18
Downloads:	13
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Unknown
Secondary title:	O natančnosti nekaterih rezultatov, ki povezujejo prereze in prekrižna števila
Secondary abstract:	It is already known that for very small edge cuts in graphs, the crossing number of the graph is at least the sum of the crossing number of (slightly augmented) components resulting from the cut. Under stronger connectivity condition in each cut component that was formalized as a graph operation called zip product, a similar result was obtained for edge cuts of any size, and a natural question was asked, whether this stronger condition is necessary. In this paper, we prove that the relaxed condition is not sufficient when the size of the cut is at least four, and we prove that the gap can grow quadratically with the cut size.
Secondary keywords:	matematika;teorija grafov;prekrižno število;šiv grafov;prerez v grafih;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	R96 (8 str.)
Volume:	ǂVol. ǂ17
Issue:	ǂno. ǂ1
Chronology:	2010
ID:	68561