Outside-Obstacle Representations with All Vertices on the Outer Face

TL;DR

This paper studies obstacle representations of graphs with a focus on outside-obstacle representations (OORs) using a single outer face obstacle, extending known results to broader classes of graphs and special configurations.

Contribution

It proves that all (partial) 2-trees admit an outside-obstacle representation, and characterizes convex OORs for certain graph classes, expanding the understanding of obstacle representations.

Findings

Every (partial) 2-tree has an outside-obstacle representation.

Characterization of convex OORs for trees and complete graphs minus a cycle.

Construction of regular OORs for outerpaths, cactus graphs, and grids.

Abstract

An obstacle representation of a graph $G$ consists of a set of polygonal obstacles and a drawing of $G$ as a visibility graph with respect to the obstacles: vertices are mapped to points and edges to straight-line segments such that each edge avoids all obstacles whereas each non-edge intersects at least one obstacle. Obstacle representations have been investigated quite intensely over the last few years. Here we focus on outside-obstacle representations (OORs) that use only one obstacle in the outer face of the drawing. It is known that every outerplanar graph admits such a representation. We strengthen this result by showing that every (partial) 2-tree has an OOR. We also consider restricted versions of OORs where the vertices of the graph form a convex polygon or even a regular polygon. We characterize when the complement of a tree and when a complete graph minus a simple cycle admits a convex OOR. We construct regular OORs for all (partial) outerpaths, cactus graphs, and grids.