  Limiting distributions are derived for the sparse connected components that
are present when a random graph on $n$ vertices has approximately $\half n$
edges. In particular, we show that such a graph consists entirely of trees,
unicyclic components, and bicyclic components with probability approaching
$\sqrt{2\over 3} \cosh\sqrt{5\over 18}\approx0.9325$ as $n\to\infty$. The
limiting probability that it consists of trees, unicyclic components, and at
most one other component is approximately 0.9957; the limiting probability that
it is planar lies between 0.987 and 0.9998. When a random graph evolves and the
number of edges passes $\half n$, its components grow in cyclic complexity
according to an interesting Markov process whose asymptotic structure is
derived. The probability that there never is more than a single component with
more edges than vertices, throughout the evolution, approaches
$5\pi/18\approx0.8727$. A ``uniform'' model of random graphs, which allows
self-loops and multiple edges, is shown to lead to formulas that are
substantially simpler than the analogous formulas for the classical random
graphs of Erd\H{o}s and R\'enyi. The notions of ``excess'' and ``deficiency,''
which are significant characteristics of the generating function as well as of
the graphs themselves, lead to a mathematically attractive structural theory
for the uniform model. A general approach to the study of stopping
configurations makes it possible to sharpen previously obtained estimates in a
uniform manner and often to obtain closed forms for the constants of interest.
Empirical results are presented to complement the analysis, indicating the
typical behavior when $n$ is near 20000.
