Unit Disk Representations of Embedded Trees, Outerplanar and Multi-Legged Graphs
Sujoy Bhore, Maarten L\"offler, Soeren Nickel, Martin N\"ollenburg

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 represents each vertex of as a unit disk in the plane, such that two disks intersect if and only if their vertices are adjacent in . 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.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Plant Surface Properties and Treatments
