String graphs. I.: the number of critical nonstring graphs is infinite
Journal of Combinatorial Theory Series B
String graphs. II.: Recognizing string graphs is NP-hard
Journal of Combinatorial Theory Series B
String graphs requiring exponential representations
Journal of Combinatorial Theory Series B
Recognizing string graphs in NP
Journal of Computer and System Sciences - STOC 2002
Journal of Computer and System Sciences - STOC 2001
Crossing number, pair-crossing number, and expansion
Journal of Combinatorial Theory Series B
Topological Graphs with No Large Grids
Graphs and Combinatorics
On the maximum number of edges in topological graphs with no four pairwise crossing edges
Proceedings of the twenty-second annual symposium on Computational geometry
How Many Ways Can One Draw A Graph?
Combinatorica
Coloring kk-free intersection graphs of geometric objects in the plane
Proceedings of the twenty-fourth annual symposium on Computational geometry
On grids in topological graphs
Proceedings of the twenty-fifth annual symposium on Computational geometry
A bipartite strengthening of the Crossing Lemma
Journal of Combinatorial Theory Series B
On the Ramsey multiplicity of complete graphs
Combinatorica
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Given a collection C of curves in the plane, its string graph is defined as the graph with vertex set C, in which two curves in C are adjacent if and only if they intersect. Given a partially ordered set (P,incomparability graph is the graph with vertex set P, in which two elements of P are adjacent if and only if they are incomparable. It is known that every incomparability graph is a string graph. For "dense" string graphs, we establish a partial converse of this statement. We prove that for every ε0 there exists δ0 with the property that if C is a collection of curves whose string graph has at least ε |C|2 edges, then one can select a subcurve γ' of each γ ∈ C such that the string graph of the collection {γ':γ ∈ C} has at least δ |C|2 edges and is an incomparability graph. We also discuss applications of this result to extremal problems for string graphs and edge intersection patterns in topological graphs.