Weak Unit Disk and Interval Representation of Planar Graphs

TL;DR

This paper investigates a specialized intersection representation of planar graphs using unit disks and intervals, proving NP-hardness for certain cases and providing positive results for specific graph classes.

Contribution

It introduces a new intersection representation problem with unit disks and intervals, proving NP-hardness, and identifies classes of graphs that always admit such representations.

Findings

NP-hard to decide representation existence for given edge-partitions

Series-parallel graphs always admit such representations

Certain planar and outerplanar graphs cannot be represented with unit intervals

Abstract

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.