The expected size of some graphs in computational geometry

Abstract

We consider n independent points with a common but arbitrary density f in R d . Two points ( X i , X j ) are joined by an edge when a certain set S( X i , X j ) does not contain any other data points. The expected number E( N) of edges in the graph depends upon n, f and the definition of S. Examples include rectangle, spheres and loons; these lead to the graph of all dominance pairs, the Gabriel graph and the relative neighborhood graph, respectively. Other graphs covered by our analysis include the nearest neighbor graph and the directional nearest neighbor graph. In all cases, we obtain asymptotic lower bounds that do not depend upon f (and are hence useful in all applications involving these graphs, since we usually do not know f). For sparse graphs, exact asymptotic constants are obtained for E( N) that are valid for all densities.