Recognizing Weighted Disk Contact Graphs

TL;DR

This paper investigates the computational complexity of recognizing weighted disk contact graphs, proving NP-hardness for certain classes and providing efficient algorithms for others, thus clarifying the boundary between hard and tractable cases.

Contribution

It extends the understanding of recognition complexity by establishing NP-hardness for specific graph classes and offering linear-time algorithms for particular cases.

Findings

NP-hardness for outerplanar graphs with unit weights

NP-hardness for stars with arbitrary weights

Linear-time recognition algorithms for caterpillars with unit weights

Abstract

Disk contact representations realize graphs by mapping vertices bijectively to interior-disjoint disks in the plane such that two disks touch each other if and only if the corresponding vertices are adjacent in the graph. Deciding whether a vertex-weighted planar graph can be realized such that the disks' radii coincide with the vertex weights is known to be NP-hard. In this work, we reduce the gap between hardness and tractability by analyzing the problem for special graph classes. We show that it remains NP-hard for outerplanar graphs with unit weights and for stars with arbitrary weights, strengthening the previous hardness results. On the positive side, we present constructive linear-time recognition algorithms for caterpillars with unit weights and for embedded stars with arbitrary weights.