Can everybody sit closer to their friends than their enemies?

Abstract

Signed graphs are graphs with signed edges. They are commonly used to represent positive and negative relationships in social networks. While balance theory and clusterizable graphs deal with signed graphs, recent empirical studies have proved that they fail to reflect some current practices in real social networks. In this paper we address the issue of drawing signed graphs and capturing such social interactions. We relax the previous assumptions to define a drawing as a model in which every vertex has to be placed closer to its neighbors connected through a positive edge than its neighbors connected through a negative edge in the resulting space. Based on this definition, we address the problem of deciding whether a given signed graph has a drawing in a given ℓ-dimensional Euclidean space. We focus on the 1-dimensional case, where we provide a polynomial time algorithm that decides if a given complete signed graph has a drawing, and provides it when applicable. © 2011 Springer-Verlag GmbH.