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

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.