Weak Unit Disk Contact Representations for Graphs without Embedding

TL;DR

This paper investigates the recognition problem of weak unit disk contact representations for specific graph classes without fixed embedding, providing efficient algorithms for caterpillars and proving NP-hardness for trees.

Contribution

It introduces a linear time recognition algorithm for caterpillars and establishes NP-hardness for trees regarding weak unit disk contact representations.

Findings

Linear time recognition for caterpillars.

NP-hardness for trees.

Insights into graph representation complexity.

Abstract

Weak unit disk contact graphs are graphs that admit representing nodes as a collection of internally disjoint unit disks whose boundaries touch if there is an edge between the corresponding nodes. In this work we focus on graphs without embedding, i.e., the neighbor order can be chosen arbitrarily. We give a linear time algorithm to recognize whether a caterpillar, a graph where every node is adjacent to or on a central path, allows a weak unit disk contact representation. On the other hand, we show that it is NP-hard to decide whether a tree allows such a representation.