Grid Obstacle Representation of Graphs

TL;DR

This paper introduces grid obstacle representations for graphs, showing planar graphs can be represented in 2D, non-planar graphs cannot always, but all graphs can be represented in 3D, and establishing NP-hardness of embedding such representations.

Contribution

It proves that planar graphs have grid obstacle representations in 2D, non-planar graphs do not always, all graphs in 3D, and demonstrates NP-hardness of embedding these representations.

Findings

Planar graphs admit 2D grid obstacle representations.

Some non-planar graphs do not admit such representations.

All graphs admit 3D grid obstacle representations.

Abstract

The grid obstacle representation, or alternately, $\ell_1$-obstacle representation of a graph $G=(V,E)$ is an injective function $f:V \rightarrow \mathbb{Z}^2$ and a set of point obstacles $\mathcal{O}$ on the grid points of $\mathbb{Z}^2$ (where no vertex of $V$ has been mapped) such that $uv$ is an edge in $G$ if and only if there exists a Manhattan path between $f(u)$ and $f(v)$ in $\mathbb{Z}^2$ avoiding the obstacles of $\mathcal{O}$ and points in $f(V)$. This work shows that planar graphs admit such a representation while there exist some non-planar graphs that do not admit such a representation. Moreover, we show that every graph admits a grid obstacle representation in $\mathbb{Z}^3$. We also show NP-hardness result for the point set embeddability of an $\ell_1$-obstacle representation.