Construction of crossing-critical graphs
Discrete Mathematics
Journal of Combinatorial Theory Series B
Minimal graphs with crossing number at least k
Journal of Combinatorial Theory Series B
Graph Drawing: Algorithms for the Visualization of Graphs
Graph Drawing: Algorithms for the Visualization of Graphs
Crossing-Critical Graphs and Path-Width
GD '01 Revised Papers from the 9th International Symposium on Graph Drawing
Which Crossing Number is it, Anyway?
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
European Journal of Combinatorics - Special issue: Topological graph theory
Improved upper bounds on the crossing number
Proceedings of the twenty-fourth annual symposium on Computational geometry
Note: Crossing-critical graphs with large maximum degree
Journal of Combinatorial Theory Series B
Nested cycles in large triangulations and crossing-critical graphs
Journal of Combinatorial Theory Series B
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The crossing number of a graph G, denoted by cr(G), is defined as 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)G) for all edges e of G. We prove that crossing-critical graphs have "bounded path-width" (by a function of the crossing number), which roughly means that such graphs are made up of small pieces joined in a linear way on small cut-sets. Equivalently, a crossing-critical graph cannot contain a subdivision of a "large" binary tree. This assertion was conjectured earlier by Salazar (J. Geelen, B. Richter, G. Salazar, Embedding grids on surfaces, manuscript, 2000).