Geodesic Obstacle Representation of Graphs
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 straightline 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 straightline 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 straightline 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 obstaclesavoiding 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.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1803.03705
 Bibcode:
 2018arXiv180303705B
 Keywords:

 Computer Science  Computational Geometry