Outside-Obstacle Representations with All Vertices on the Outer Face

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 that use only one obstacle in the outer face of the drawing. It is known that every outerplanar graph admits such a representation [Alpert, Koch, Laison; DCG 2010]. We strengthen this result by showing that every partial 2-tree has an outside-obstacle representation. We also consider a restricted version of outside-obstacle representations where the vertices lie on a regular polygon. We construct such regular representations for partial outerpaths, partial cactus graphs, and partial grids.

Bounding and Computing Obstacle Numbers of Graphs

“…Analogous hardness results hold with respect to inside and general obstacles. Every partial 2-tree has an outside-obstacle representation [10]. For (partial) outerpaths, cactus graphs, and grids, it is possible to construct outside-obstacle representations where the vertices are those of a regular polygon [10].…”