Pseudoline arrangement graphs: degree sequences and eccentricities

TL;DR

This paper investigates pseudoline arrangement graphs, providing a degree sequence criterion for their realization and analyzing vertex eccentricities, including diameter and diametrical vertices, in such graphs.

Contribution

It introduces a simple degree sequence criterion for recognizing pseudoline arrangement graphs and characterizes their diametrical vertices and diameter.

Findings

Diameter of pseudoline arrangement graphs is n-2.

A degree sequence criterion determines realizability.

Diametrical vertices are characterized.

Abstract

A pseudoline arrangement graph is a planar graph induced by an embedding of a (simple) pseudoline arrangement. We study the corresponding graph realization problem and properties of pseudoline arrangement graphs. In the first part, we give a simple criterion based on the degree sequence that says whether a degree sequence will have a pseudoline arrangement graph as one of its realizations. In the second part, we study the eccentricities of vertices in such graphs. We observe that the diameter (maximum eccentricity of a vertex in the graph) of any pseudoline arrangement graph on $n$ pseudolines is $n-2$. Then we characterize the diametrical vertices (whose eccentricity is equal to the graph diameter) of pseudoline arrangement graphs. These results hold for line arrangement graphs as well.