String graphs and incomparability graphs

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,<), its 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 ε>0 there exists δ>0 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.

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