Characterizing all graphs with 2-exceptional edges

Abstract

Dirac and Shuster in 1954 exhibited a simple proof of Kuratowski theorem by showing that any 1-crossing-critical edge of G belongs to a Kuratowski subdivision of G. In 1983, Širáň extended this result to any 2-crossing-critical edge e with endvertices b and c of a graph G with crossing number at least two, whenever no two blocks of G − b − c contain all its vertices. Calling an edge f of G k-exceptional whenever f is k-crossing-critical and it does not belong to any Kuratowski subgraph of G, he showed that simple 3-connected graphs with k-exceptional edges exist for any k ≥ 6, and they exist even for arbitrarily large difference of cr(G) − cr(G − f). In 1991, Kochol constructed such examples for any k ≥ 4, and commented that Širáň’s result holds for any simple graph.

Examining the case when two blocks contain all the vertices of G − b − c, we show that graphs with k-exceptional edges exist for any k ≥ 2, albeit not necessarily simple. We confirm that no such simple graphs with 2-exceptional edges exist by applying the techniques of the recent characterization of 2-crossing-critical graphs to explicitly describe the set of all graphs with 2-exceptional edges and noting they all contain parallel edges. In this context, the paper can be read as an accessible prelude to the characterization of 2-crossing-critical graphs.