On 2-crossing-critical graphs with a V8-minor
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Date
2014-05-22
Authors
Arroyo Guevara, Alan Marcelo
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Journal ISSN
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Publisher
University of Waterloo
Abstract
The crossing number of a graph is the minimum number of pairwise edge crossings in a drawing of a graph. A graph $G$ is $k$-crossing-critical if it has crossing number at least $k$, and any subgraph of $G$ has crossing number less than $k$. A consequence of Kuratowski's theorem is that 1-critical graphs are subdivisions of $K_{3,3}$ and $K_{5}$. The graph $V_{2n}$ is a $2n$-cycle with $n$ diameters. Bokal, Oporowski,
Richter and Salazar found in \cite{bigpaper} all the critical graphs except the ones that contain a $V_{8}$ minor and no $V_{10}$ minor.
We show that a 4-connected graph $G$ has crossing number at least 2 if and only if for each pair of disjoint edges there are two disjoint cycles containing them. Using a generalization of this result we found limitations for the 2-crossing-critical graphs remaining to classify. We showed that peripherally 4-connected 2-crossing-critical graphs have at most 4001 vertices. Furthermore, most 3-connected 2-crossing-critical graphs are obtainable by small modifications of the peripherally 4-connected ones.
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Keywords
graph theory, crossing numbers, disjoint paths, crossing critical