What graphs are 2-dot product graphs?

Abstract

Let d ≥ 1 be an integer. From a set of d-dimensional vectors, we obtain a d-dot product graph by letting each vector a u correspond to a vertex u and by adding an edge between two vertices u and v if and only if their dot product a u · a v ≥ t, for some fixed, positive threshold t. Dot product graphs can be used to model social networks. Recognizing a d-dot product graph is known to be NP-hard for all fixed d ≥ 2. To understand the position of d-dot product graphs in the landscape of graph classes, we consider the case d = 2, and investigate how 2-dot product graphs relate to a number of other known graph classes including a number of well-known classes of intersection graphs.