Browsing by Author "Austin, Beth Ann"

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2-crossing critical graphs with a V8 minor
(University of Waterloo, 2012-01-17T20:51:50Z) Austin, Beth Ann
The crossing number of a graph is the minimum number of pairwise crossings of edges among all planar drawings of the graph. A graph G is k-crossing critical if it has crossing number k and any proper subgraph of G has a crossing number less than k. The set of 1-crossing critical graphs is is determined by Kuratowski’s Theorem to be {K5, K3,3}. Work has been done to approach the problem of classifying all 2-crossing critical graphs. The graph V2n is a cycle on 2n vertices with n intersecting chords. The only remaining graphs to find in the classification of 2-crossing critical graphs are those that are 3-connected with a V8 minor but no V10 minor. This paper seeks to fill some of this gap by defining and completely describing a class of graphs called fully covered. In addition, we examine other ways in which graphs may be 2-crossing critical. This discussion classifies all known examples of 3-connected, 2-crossing critical graphs with a V8 minor but no V10 minor.