Minor crossing number is additive over arbitrary cuts

Details Minor crossing number is additive over arbitrary cuts Minor crossing number is additive over arbitrary cuts

2011-11-25

arxiv.org/abs/1111.6024v1

Mathematics

Combinatorics

11 pages

Scientific paper

We prove that if $G$ is a graph with an minimal edge cut $F$ of size three and $G_1$, $G_2$ are the two (augmented) components of $G-F$, then the crossing number of $G$ is equal to the sum of crossing numbers of $G_1$ and $G_2$. Combining with known results, this implies that crossing number is additive over edge-cuts of size $d$ for $d\in\{0, 1, 2, 3\}$, whereas there are counterexamples for every $d\ge 4$. 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.

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