Before revision:  such that uv is an edge in G if and only if there exists a Manhattan path between f(u) and f(v) in Z^2 avoiding the obstacles of O . The grid obstacle number of a graph is the smallest number of obstacles needed for the grid obstacle representation of G. This work shows that planar graphs admit such a representation while there exists some non-planar graphs that do not admit such a representation.
After revision:  such that uv is an edge in G if and only if there exists a Manhattan path between f(u) and f(v) in Z^2 avoiding the obstacles of O . The grid obstacle number of a graph is the smallest number of obstacles needed for the grid obstacle representation of G. This work shows that planar graphs admit such a representation while there exist some non-planar graphs that do not admit such a representation.