  When studying networks using random graph models, one is sometimes faced with
situations where the notion of adjacency between nodes reflects multiple
constraints. Traditional random graph models are insufficient to handle such
situations.
  A simple idea to account for multiple constraints consists in taking the
intersection of random graphs. In this paper we initiate the study of random
graphs so obtained through a simple example. We examine the intersection of an
Erdos-Renyi graph and of one-dimensional geometric random graphs. We
investigate the zero-one laws for the property that there are no isolated
nodes. When the geometric component is defined on the unit circle, a full
zero-one law is established and we determine its critical scaling. When the
geometric component lies in the unit interval, there is a gap in that the
obtained zero and one laws are found to express deviations from different
critical scalings. In particular, the first moment method requires a larger
critical 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 the
absence of isolated nodes are identical in the geometric random graphs on both
the unit interval and unit circle.
