Unit Disk Representations of Embedded Trees, Outerplanar and Multi-Legged Graphs

TL;DR

This paper investigates the computational complexity of representing various classes of graphs with unit disks, proving NP-hardness for some and providing efficient algorithms for specific subclasses like caterpillars and lobsters.

Contribution

It establishes NP-hardness results for outerplanar graphs and embedded trees, and offers linear-time algorithms for caterpillar and lobster graphs regarding unit disk representations.

Findings

NP-hard to decide UDR for outerplanar graphs and embedded trees

Linear-time characterization for caterpillar graphs with UDR

Linear-time decision for weak UDC in lobster graphs

Abstract

A unit disk intersection representation (UDR) of a graph $G$ represents each vertex of $G$ as a unit disk in the plane, such that two disks intersect if and only if their vertices are adjacent in $G$. A UDR with interior-disjoint disks is called a unit disk contact representation (UDC). We prove that it is NP-hard to decide if an outerplanar graph or an embedded tree admits a UDR. We further provide a linear-time decidable characterization of caterpillar graphs that admit a UDR. Finally we show that it can be decided in linear time if a lobster graph admits a weak UDC, which permits intersections between disks of non-adjacent vertices.