Intersection Graphs of Pseudosegments: Chordal Graphs

TL;DR

This paper explores which chordal graphs can be represented as intersection graphs of pseudosegments, providing constructions, counterexamples, and combinatorial bounds related to pseudosegment representations.

Contribution

It characterizes the representability of chordal graphs as pseudosegment intersection graphs and introduces bounds on the number of k-segments in pseudoline arrangements.

Findings

All chordal graphs representable as subpath intersection graphs on a tree are pseudosegment intersection graphs.

Certain intersection graphs of substars of a star are not representable as pseudosegment intersection graphs.

The number of k-segments in pseudoline arrangements is bounded by O(n^2) for fixed k.

Abstract

We investigate which chordal graphs have a representation as intersection graphs of pseudosegments. For positive we have a construction which shows that all chordal graphs that can be represented as intersection graph of subpaths on a tree are pseudosegment intersection graphs. We then study the limits of representability. We describe a family of intersection graphs of substars of a star which is not representable as intersection graph of pseudosegments. The degree of the substars in this example, however, has to get large. A more intricate analysis involving a Ramsey argument shows that even in the class of intersection graphs of substars of degree three of a star there are graphs that are not representable as intersection graph of pseudosegments. Motivated by representability questions for chordal graphs we consider how many combinatorially different k-segments, i.e., curves crossing k distinct lines, an arrangement of n pseudolines can host. We show that for fixed k this number is in O(n^2). This result is based on a k-zone theorem for arrangements of pseudolines that should be of independent interest.