Bipartite dot product graphs

Abstract

Given a bipartite graph $G=(X,Y,E)$, the bipartite dot product representation of G is a function $f:X\cup Y\to {\mathbb{R}}^k$ and a positive threshold t such that for any $x\in X$ and $y\in Y$, $xy\in E$ if and only if $f(x)\cdot f(y)\ge t$. The minimum k such that a bipartite dot product representation exists for G is the bipartite dot product dimension of G, denoted $bdp(G)$. We will show that such representations exist for all bipartite graphs as well as give an upper bound for the bipartite dot product dimension of any graph. We will also characterize the bipartite graphs of bipartite dot product dimension 1 by their forbidden subgraphs.

Keywords:

• Graph theory
• dot product graphs
• bipartite graphs
• graph representations
• subgraph categorization

AMS CLASSIFICATIONS:

• 05C62
• 05C75

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 This paper is derived from the first author's Ph.D. thesis under the supervision of the second author.