Graphs with Plane Outside-Obstacle Representations

TL;DR

This paper characterizes and recognizes plane outside-obstacle graph representations, where obstacles are outside the graph's unbounded face, providing a linear-time recognition algorithm and linking these graphs to known classes.

Contribution

It provides a combinatorial characterization and a linear-time recognition algorithm for plane outside-obstacle graphs, a new class generalizing classical visibility graphs.

Findings

Characterization of biconnected graphs with outside-obstacle representations

Linear-time recognition algorithm for these graphs

Plane vertex-polygon visibility graphs are exactly maximal outerplanar graphs

Abstract

An \emph{obstacle representation} of a graph consists of a set of polygonal obstacles and a distinct point for each vertex such that two points see each other if and only if the corresponding vertices are adjacent. Obstacle representations are a recent generalization of classical polygon--vertex visibility graphs, for which the characterization and recognition problems are long-standing open questions. In this paper, we study \emph{plane outside-obstacle representations}, where all obstacles lie in the unbounded face of the representation and no two visibility segments cross. We give a combinatorial characterization of the biconnected graphs that admit such a representation. Based on this characterization, we present a simple linear-time recognition algorithm for these graphs. As a side result, we show that the plane vertex--polygon visibility graphs are exactly the maximal outerplanar graphs and that every chordal outerplanar graph has an outside-obstacle representation.