Geodesic Obstacle Representation of Graphs

TL;DR

This paper introduces geodesic obstacle representations of graphs, generalizing existing models by considering shortest paths in various metric spaces, including Euclidean and graph metrics, to better understand graph embeddings with obstacle constraints.

Contribution

It extends obstacle representation models to include geodesic shortest paths in Euclidean and other metric spaces, unifying and generalizing previous obstacle and grid obstacle representations.

Findings

Generalizes obstacle representations to geodesic shortest paths

Unifies models across Euclidean, polyhedral, and graph metrics

Raises new questions on graph embedding properties

Abstract

An obstacle representation of a graph is a mapping of the vertices onto points in the plane and a set of connected regions of the plane (called obstacles) such that the straight-line segment connecting the points corresponding to two vertices does not intersect any obstacles if and only if the vertices are adjacent in the graph. The obstacle representation and its plane variant (in which the resulting representation is a plane straight-line embedding of the graph) have been extensively studied with the main objective of minimizing the number of obstacles. Recently, Biedl and Mehrabi (GD 2017) studied grid obstacle representations of graphs in which the vertices of the graph are mapped onto the points in the plane while the straight-line segments representing the adjacency between the vertices is replaced by the $L_1$ (Manhattan) shortest paths in the plane that avoid obstacles. In this paper, we introduce the notion of geodesic obstacle representations of graphs with the main goal of providing a generalized model, which comes naturally when viewing line segments as shortest paths in the Euclidean plane. To this end, we extend the definition of obstacle representation by allowing some obstacles-avoiding shortest path between the corresponding points in the underlying metric space whenever the vertices are adjacent in the graph. We consider both general and plane variants of geodesic obstacle representations (in a similar sense to obstacle representations) under any polyhedral distance function in $\mathbb{R}^d$ as well as shortest path distances in graphs. Our results generalize and unify the notions of obstacle representations, plane obstacle representations and grid obstacle representations, leading to a number of questions on such embeddings.