The Dimension of Valid Distance Drawings of Signed Graphs

Abstract

A signed graph is an undirected graph with a sign assignment to its edges. Drawing a signed graph in $\mathbb {R}^k$ means finding an injection of the set of nodes into $\mathbb {R}^k$. A valid distance drawing of a signed graph in $\mathbb {R}^k$ is an injection of the nodes into $\mathbb {R}^k$ such that, for each node, all its positive neighbors are closer than its negative neighbors. This work addresses the problem of finding L(n), the smallest dimension such that any graph on n nodes has a valid distance drawing in a Euclidean space of dimension L(n). We show that $\lfloor \log _5(n-3) \rfloor + 1 \le L(n) \le n-2$. We also compute exact values for L(n) up to $n=7$.