Outside-Obstacle Representations with All Vertices on the Outer Face

An \emph{obstacle representation} of a graph $G$ consists of a set of polygonal obstacles and a drawing of $G$ as a \emph{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 \emph{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.