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When studying networks using random graph models, one is sometimes faced withsituations where the notion of adjacency between nodes reflects multipleconstraints. Traditional random graph models are insufficient to handle suchsituations.  A simple idea to account for multiple constraints consists in taking theintersection of random graphs. In this paper we initiate the study of randomgraphs so obtained through a simple example. We examine the intersection of anErdos-Renyi graph and of one-dimensional geometric random graphs. Weinvestigate the zero-one laws for the property that there are no isolatednodes. When the geometric component is defined on the unit circle, a fullzero-one law is established and we determine its critical scaling. When thegeometric component lies in the unit interval, there is a gap in that theobtained zero and one laws are found to express deviations from differentcritical scalings. In particular, the first moment method requires a largercritical scaling than in the unit circle case in order to obtain the one law.This discrepancy is somewhat surprising given that the zero-one laws for theabsence of isolated nodes are identical in the geometric random graphs on boththe unit interval and unit circle.

Intersecting random graphs and networks with multiple adjacency constraints: A simple example