Crossing number additivity over edge cuts

Drago Bokal (Author), Markus Chimani (Author), Jesús Leanõs (Author)

Abstract

Consider a graph ▫$G$▫ with a minimal edge cut ▫$F$▫ and let ▫$G_1$▫, ▫$G_2$▫ be the two (augmented) components of ▫$G-F$▫. A long-open question asks under which conditions the crossing number of ▫$G$▫ is (greater than or) equal to the sum ofcthe crossing numbers of ▫$G_1$▫ and ▫$G_2$▫ - which would allow us to consider those graphs separately. It is known that crossing number is additive for ▫$|F| \in \{0,1,2\}$▫ and that there exist graphs violating this property with ▫$|F| \ge 4$▫. In this paper, we show that crossing number is additive for ▫$|F|=3$▫, thus closing the final gap in the question. The techniques generalize to show that minor crossing number is additive over edge cuts of arbitrary size, as well as to provide bounds for crossing number additivity in arbitrary surfaces. We point out several applications to exact crossing number computation and crossing-critical graphs, as well as provide a very general lower bound for the minor crossing number of the Cartesian product of an arbitrary graph with a tree.

Keywords

matematika;teorija grafov;prekrižno število;minor;mathematics;graph theory;crossing number;

Data

Language:	English
Year of publishing:	2013
Typology:	1.01 - Original Scientific Article
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
UDC:	519.17
COBISS:	16624473
ISSN:	0195-6698
Views:	35
Downloads:	22
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary keywords:	matematika;teorija grafov;prekrižno število;minor;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 1010-1018
Volume:	Vol. 34
Issue:	iss. 6
Chronology:	2013
ID:	1476829