## String graphs and incomparability graphs

##### Author(s)

Fox, Jacob; Pach, Janos
DownloadFox_String graphs.pdf (332.1Kb)

PUBLISHER_CC

# Publisher with Creative Commons License

Creative Commons Attribution

##### Terms of use

##### Metadata

Show full item record##### Abstract

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|[superscript 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|[superscript 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.

##### Date issued

2012-04##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Advances in Mathematics

##### Publisher

Elsevier

##### Citation

Fox, Jacob, and Janos Pach. “String Graphs and Incomparability Graphs.” Advances in Mathematics 230, no. 3 (June 2012): 1381–1401.

Version: Author's final manuscript

##### ISSN

00018708

1090-2082