  We present an interactive framework that, given a membership test for a graph
class $\mathcal{G}$ and a number $k$, finds and tests unavoidable sets for the
class of graphs in $\mathcal{G}$ of path-width at most $k$. We put special
emphasis on the case that $\mathcal{G}$ is the class of cubic graphs and tailor
the algorithm to this case. In particular, we introduce the new concept of
high-degree-first path-decompositions, which yields highly efficient pruning
techniques.
  Using this framework we determine all extremal girth values of cubic graphs
of path-width $k$ for all $k \in \{3,\dots, 10\}$. Moreover, we determine all
smallest graphs which take on these extremal girth values. As a further
application of our framework we characterise the extremal cubic graphs of
path-width 3 and girth 4.
