Computational Complexity Aspects of 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.