Crossing-critical graphs and path-width

Abstract

The crossing number cr(G) of a graph G, is the smallest possible number of edge-crossings in a drawing of G in the plane. A graph G is crossing-critical if cr(G - e) < cr(G) for all edges e of G. G. Salazar conjectured in 1999 that crossing-critical graphs have pathwidth bounded by a function of their crossing number, which roughly means that such graphs are made up of small pieces joined in a linear way on small cut-sets. That conjecture was recently proved by the author [9]. Our paper presents that result together with a brief sketch of proof ideas. The main focus of the paper is on presenting a new construction of crossing-critical graphs, which, in particular, gives a nontrivial lower bound on the path-width. Our construction may be interesting also to other areas concerned with the crossing number. © Springer-Verlag Berlin Heidelberg 2002.