dc.contributor.author Arroyo Guevara, Alan Marcelo dc.date.accessioned 2014-05-22 17:51:23 (GMT) dc.date.available 2014-05-22 17:51:23 (GMT) dc.date.issued 2014-05-22 dc.date.submitted 2014-05-20 dc.identifier.uri http://hdl.handle.net/10012/8494 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, en 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. dc.language.iso en en 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.title On 2-crossing-critical graphs with a V8-minor en dc.type Master Thesis en dc.pending false dc.subject.program Combinatorics and Optimization en uws-etd.degree.department Combinatorics and Optimization en uws-etd.degree Master of Mathematics en uws.typeOfResource Text en uws.peerReviewStatus Unreviewed en uws.scholarLevel Graduate en
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