Weak Unit Disk and Interval Representation of Graphs

We study a variant of intersection representations with unit balls: 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 edges, the goal is to represent the vertices of the graph by unit-size balls so that the balls for 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, we show that series-parallel graphs (which include outerplanar graphs) admit such a representation with unit disks for any near/far bipartition of the edges. The unit-interval on the line variant is equivalent to threshold graph coloring, in which context it is known that there exist girth-3 planar graphs (even outerplanar graphs) that do not admit such coloring. We extend this result to girth-4 planar graphs. On the other hand, we show that all triangle-free outerplanar graphs and all planar graphs with maximum average degree less than 26/11 have such a coloring, via unit-interval intersection representation on the line. This gives a simple proof that all planar graphs with girth at least 13 have a unit-interval intersection representation on the line.