Analyzing Tree Attachments in 2-Crossing-Critical Graphs with a V8 Minor

Abstract

The crossing number of a graph is the minimum number of pairwise edge crossings in a drawing of the graph in the plane. A graph G is k-crossing-critical if its crossing number is at least k and if every proper subgraph H of G has crossing number less than k.

It follows directly from Kuratowski's Theorem that the 1-crossing-critical graphs are precisely the subdivisions of K{3,3} and K5. Characterizing the 2-crossing-critical graphs is an interesting open problem.

Much progress has been made in characterizing the 2-crossing-critical graphs. The only remaining unexplained such graphs are those which are 3-connected, have a V8 minor but no V10 minor, and embed in the real projective plane. This thesis seeks to extend previous attempts at classifying this particular set of graphs by examining the graphs in this category where a tree structure is attached to a subdivision of V8.

In this paper, we analyze which of the 106 possible 3-stars can be attached to a subdivision H of V8 in a 3-connected 2-crossing-critical graph. This analysis leads to a strong result, where we demonstrate that if a k-star is attached to a V8 in a 2-crossing-critical graph, then k <= 4. Finally, we significantly restrict the remaining trees which still need to be investigated under the same conditions.