  Given a set $\xi=\{H_1,H_2,...\}$ of connected non acyclic graphs, a
$\xi$-free graph is one which does not contain any member of $% \xi$ as copy.
Define the excess of a graph as the difference between its number of edges and
its number of vertices. Let ${\gr{W}}_{k,\xi}$ be theexponential generating
function (EGF for brief) of connected $\xi$-free graphs of excess equal to $k$
($k \geq 1$). For each fixed $\xi$, a fundamental differential recurrence
satisfied by the EGFs ${\gr{W}}_{k,\xi}$ is derived. We give methods on how to
solve this nonlinear recurrence for the first few values of $k$ by means of
graph surgery. We also show that for any finite collection $\xi$ of non-acyclic
graphs, the EGFs ${\gr{W}}_{k,\xi}$ are always rational functions of the
generating function, $T$, of Cayley's rooted (non-planar) labelled trees. From
this, we prove that almost all connected graphs with $n$ nodes and $n+k$ edges
are $\xi$-free, whenever $k=o(n^{1/3})$ and $|\xi| < \infty$ by means of
Wright's inequalities and saddle point method. Limiting distributions are
derived for sparse connected $\xi$-free components that are present when a
random graph on $n$ nodes has approximately $\frac{n}{2}$ edges. In particular,
the probability distribution that it consists of trees, unicyclic components,
$...$, $(q+1)$-cyclic components all $\xi$-free is derived. Similar results are
also obtained for multigraphs, which are graphs where self-loops and
multiple-edges are allowed.
