Given a set $��=\{H_1,H_2,...\}$ of connected non acyclic graphs, a $��$-free graph is one which does not contain any member of $% ��$ as copy. Define the excess of a graph as the difference between its number of edges and its number of vertices. Let ${\gr{W}}_{k,��}$ be theexponential generating function (EGF for brief) of connected $��$-free graphs of excess equal to $k$ ($k \geq 1$). For each fixed $��$, a fundamental differential recurrence satisfied by the EGFs ${\gr{W}}_{k,��}$ 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 $��$ of non-acyclic graphs, the EGFs ${\gr{W}}_{k,��}$ 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 $��$-free, whenever $k=o(n^{1/3})$ and $|��| < \infty$ by means of Wright's inequalities and saddle point method. Limiting distributions are derived for sparse connected $��$-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 $��$-free is derived. Similar results are also obtained for multigraphs, which are graphs where self-loops and multiple-edges are allowed.