  We discuss a graph-based approach for testing spatial point patterns. This
approach falls under the category of data-random graphs, which have been
introduced and used for statistical pattern recognition in recent years. Our
goal is to test complete spatial randomness against segregation and association
between two or more classes of points. To attain this goal, we use a particular
type of parametrized random digraph called proximity catch digraph (PCD) which
is based based on relative positions of the data points from various classes.
The statistic we employ is the relative density of the PCD. When scaled
properly, the relative density of the PCD is a $U$-statistic. We derive the
asymptotic distribution of the relative density, using the standard central
limit theory of $U$-statistics. The finite sample performance of the test
statistic is evaluated by Monte Carlo simulations, and the asymptotic
performance is assessed via Pitman's asymptotic efficiency, thereby yielding
the optimal parameters for testing. Furthermore, the methodology discussed in
this article is also valid for data in multiple dimensions.
