The use of data-random graphs in statistical testing of spatial patterns is introduced recently. In this approach, a random directed graph is constructed from the data using the relative positions of the points from various classes. Different random graphs result from different definitions of the proximity region associated with each data point and different graph statistics can be employed for pattern testing. The approach used in this article is based on underlying graphs of a family of data-random digraphs which is determined by a family of parameterized proximity maps. The relative edge density of the AND- and OR-underlying graphs is used as the summary statistic, providing an alternative to the relative arc density and domination number of the digraph employed previously. Properly scaled, relative edge density of the underlying graphs is a U-statistic, facilitating analytic study of its asymptotic distribution using standard U-statistic central limit theory. The approach is illustrated with an application to the testing of bivariate spatial clustering patterns of segregation and association. Knowledge of the asymptotic distribution allows evaluation of the Pitman asymptotic efficiency, hence selection of the proximity map parameter to optimize efficiency. Asymptotic efficiency and Monte Carlo simulation analysis indicate that the AND-underlying version is better (in terms of power and efficiency) for the segregation alternative, while the OR-underlying version is better for the association alternative. The approach presented here is also valid for data in higher dimensions.