On the computational complexity of degenerate unit distance representations of graphs

TL;DR

This paper investigates the complexity of representing graphs in Euclidean space with unit distances, focusing on degenerate cases where vertices coincide, and proves that certain related decision problems are NP-hard.

Contribution

It establishes NP-hardness results for deciding if a graph can be homomorphically mapped to a graph with a given Euclidean dimension of at least 2.

Findings

Deciding homomorphism to graphs with dimension ≥ 2 is NP-hard.

Determining the Euclidean dimension of a graph is computationally difficult.

Degenerate representations involve vertices coinciding, adding complexity to the problem.

Abstract

Some graphs admit drawings in the Euclidean k-space in such a (natu- ral) way, that edges are represented as line segments of unit length. Such drawings will be called k dimensional unit distance representations. When two non-adjacent vertices are drawn in the same point, we say that the representation is degenerate. The dimension (the Euclidean dimension) of a graph is defined to be the minimum integer k needed that a given graph has non-degenerate k dimensional unit distance representation (with the property that non-adjacent vertices are mapped to points, that are not distance one appart). It is proved that deciding if an input graph is homomorphic to a graph with dimension k >= 2 (with the Euclidean dimension k >= 2) are NP-hard problems.