On 2-crossing-critical graphs with a V8-minor
| dc.contributor.author | Arroyo Guevara, Alan Marcelo | |
| dc.date.accessioned | 2014-05-22T17:51:23Z | |
| dc.date.available | 2014-05-22T17:51:23Z | |
| dc.date.issued | 2014-05-22 | |
| dc.date.submitted | 2014-05-20 | |
| dc.description.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. | en |
| dc.identifier.uri | http://hdl.handle.net/10012/8494 | |
| dc.language.iso | en | en |
| dc.pending | false | |
| dc.publisher | University of Waterloo | en |
| dc.subject | graph theory | en |
| dc.subject | crossing numbers | en |
| dc.subject | disjoint paths | en |
| dc.subject | crossing critical | en |
| dc.subject.program | Combinatorics and Optimization | en |
| dc.title | On 2-crossing-critical graphs with a V8-minor | en |
| dc.type | Master Thesis | en |
| uws-etd.degree | Master of Mathematics | en |
| uws-etd.degree.department | Combinatorics and Optimization | en |
| uws.peerReviewStatus | Unreviewed | en |
| uws.scholarLevel | Graduate | en |
| uws.typeOfResource | Text | en |