  We study a variant of intersection representations with unit balls, that is,
unit disks in the plane and unit intervals on the line. Given a planar graph
and a bipartition of the edges of the graph into near and far sets, the goal is
to represent the vertices of the graph by unit balls so that the balls
representing two adjacent vertices intersect if and only if the corresponding
edge is near. We consider the problem in the plane and prove that it is NP-hard
to decide whether such a representation exists for a given edge-partition. On
the other hand, every series-parallel graph admits such a representation with
unit disks for any near/far labeling of the edges. We also show that the
representation problem on the line is equivalent to a variant of a graph
coloring. We give examples of girth-4 planar and girth-3 outerplanar graphs
that have no such representation with unit intervals. On the other hand, all
triangle-free outerplanar graphs and all graphs with maximum average degree
less than 26/11 can always be represented. In particular, this gives a simple
proof of representability of all planar graphs with large girth.
