  Let each point of a homogeneous Poisson process on $\RR$ independently be
equipped with a random number of stubs (half-edges) according to a given
probability distribution $\mu$ on the positive integers. We consider schemes
based on Gale-Shapley stable marriage for perfectly matching the stubs to
obtain a simple graph with degree distribution $\mu$. We prove results on the
existence of an infinite component and on the length of the edges, with focus
on the case $\mu(\{2\})=1$. In this case, for the random direction stable
matching scheme introduced by Deijfen and Meester we prove that there is no
infinite component, while for the stable matching of Deijfen, H\"aggstr\"om and
Holroyd we prove that existence of an infinite component follows from a certain
statement involving a {\em finite} interval, which is overwhelmingly supported
by simulation evidence.
