Computational Complexity Aspects of Point Visibility Graphs

TL;DR

This paper investigates the computational complexity of various classic graph problems on point visibility graphs, establishing NP-hardness results and exploring the complexity of the Dominating Set problem on grid-based point visibility graphs.

Contribution

It proves NP-hardness for several problems on point visibility graphs and analyzes the complexity of Dominating Set on grid points, advancing understanding of their computational difficulty.

Findings

NP-hardness for Feedback Vertex Set, Longest Induced Path, Bisection, and F-free Vertex Deletion.

Complexity analysis of Dominating Set on grid point visibility graphs.

Insights into the computational challenges of problems on point visibility graphs.

Abstract

A point visibility graph is a graph induced by a set of points in the plane where the vertices of the graph represent the points in the point set and two vertices are adjacent if and only if no other point from the point set lies on the line segment between the two corresponding points. The set of all point visibility graphs form a graph class which is examined from a computational complexity perspective in this paper. We show NP-hardness for several classic graph problems on point visibility graphs such as Feedback Vertex Set, Longest Induced Path, Bisection and $\mathcal{F}$-free Vertex Deletion (for certain sets $\mathcal{F}$). Furthermore, we consider the complexity of the Dominating Set problem on point visibility graphs of points on a grid.