On the connectivity of visibility graphs

TL;DR

This paper investigates the connectivity properties of visibility graphs, establishing bounds on their edge- and vertex-connectivity, and demonstrating that these bounds are tight, with implications for understanding their structural robustness.

Contribution

The paper provides new bounds on the connectivity of visibility graphs, including tight bounds on vertex-connectivity based on collinearity and minimum degree, extending previous results.

Findings

Visibility graphs have diameter at most 2 unless all points are collinear.

Edge-connectivity equals minimum degree unless all vertices are collinear.

Vertex-connectivity is at least (n-1)/(l-1), tight, and at least half the minimum degree, improved to two-thirds when l=4.

Abstract

The visibility graph of a finite set of points in the plane has the points as vertices and an edge between two vertices if the line segment between them contains no other points. This paper establishes bounds on the edge- and vertex-connectivity of visibility graphs. Unless all its vertices are collinear, a visibility graph has diameter at most 2, and so it follows by a result of Plesn\'ik (1975) that its edge-connectivity equals its minimum degree. We strengthen the result of Plesn\'ik by showing that for any two vertices v and w in a graph of diameter 2, if deg(v) <= deg(w) then there exist deg(v) edge-disjoint vw-paths of length at most 4. Furthermore, we find that in visibility graphs every minimum edge cut is the set of edges incident to a vertex of minimum degree. For vertex-connectivity, we prove that every visibility graph with n vertices and at most l collinear vertices has connectivity at least (n-1)/(l-1), which is tight. We also prove the qualitatively stronger result that the vertex-connectivity is at least half the minimum degree. Finally, in the case that l=4 we improve this bound to two thirds of the minimum degree.