On complexity of mutlidistance graph recognition in $\mathbb{R}^1$

TL;DR

This paper investigates the computational complexity of recognizing graphs that can be embedded in one-dimensional space with edge lengths from a specified set, classifying all such sets based on complexity in various problem variants.

Contribution

It provides a comprehensive classification of the complexity for recognizing $ ext{A}$-embeddable graphs in $ ext{R}^1$ across different finite sets $ ext{A}$ and problem variations.

Findings

Complexity classifications for all finite sets $ ext{A}$ in $ ext{R}^1$ recognition problems.

Identification of cases where recognition is polynomial-time solvable.

Identification of cases where recognition is NP-hard.

Abstract

Let $\mathcal{A}$ be a set of positive numbers. A graph $G$ is called an $\mathcal{A}$-embeddable graph in $\mathbb{R}^d$ if the vertices of $G$ can be positioned in $\mathbb{R}^d$ so that the distance between endpoints of any edge is an element of $\mathcal{A}$. We consider the computational problem of recognizing $\mathcal{A}$-embeddable graphs in $\mathbb{R}^1$ and classify all finite sets $\mathcal{A}$ by complexity of this problem in several natural variations.