Bipartite dot product graphs

Given a bipartite graph G = (X, Y, E), the bipartite dot product representation of G is a function f : X boolean OR Y -> R-k and a positive threshold t such that for any x is an element of X and y is an element of Y, xy is an element of E if and only if f(x) . f(y) >= 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.