This paper investigates ultrafilters in the context of connectivity systems, defined as pairs (X,
f) where X is a finite set and f is a symmetric submodular function. Ultrafilters, essential in …

The crossing number is a popular tool in graph drawing and visualization, but there is not
really just one crossing number; there is a large family of crossing number notions of which …

Crossing Numbers of Graphs is the first book devoted to the crossing number, an
increasingly popular object of study with surprising connections. The field has matured into a …

A string graph is the intersection graph of a collection of continuous arcs in the plane. We
show that any string graph with m edges can be separated into two parts of roughly equal …

Near-Optimal Separators in String Graphs Page 1 Combinatorics, Probability and Computing
(2014) 23, 135–139. c Cambridge University Press 2013 doi:10.1017/S0963548313000400 …

" Based on a lecture series given by the authors at a satellite meeting of the 2006
International Congress of Mathematicians and on many articles written by them and their …

The intersection graph of a collection C of sets is a graph on the vertex set C, in which C 1, C
2∈ C are joined by an edge if and only if C 1∩ C 2≠ Ø. Erdös conjectured that the …

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 …

Tree decompositions of graphs are of fundamental importance in structural and algorithmic
graph theory. Planar decompositions generalise tree decompositions by allowing an …

A topological graph G is a graph drawn in the plane so that its edges are represented by
Jordan arcs. G is called simple, if any two edges have at most one point in common. It is …