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.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OWTU.10012/8494 |
| Date | 20 May 2014 |
| Creators | Arroyo Guevara, Alan Marcelo |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
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