Grid Obstacle Representation of Graphs

Abstract

The grid obstacle representation, or alternately, ${\ell }_1$-obstacle representation of a graph $G=(V,E)$ is an injective function $f:V\to {\mathbb{Z}}^2$ 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.

14 figures and 18 pages