  We give offline algorithms for processing a sequence of $2$ and $3$ edge and
vertex connectivity queries in a fully-dynamic undirected graph. While the
current best fully-dynamic online data structures for $3$-edge and $3$-vertex
connectivity require $O(n^{2/3})$ and $O(n)$ time per update, respectively, our
per-operation cost is only $O(\log n)$, optimal due to the dynamic connectivity
lower bound of Patrascu and Demaine. Our approach utilizes a divide and conquer
scheme that transforms a graph into smaller equivalents that preserve
connectivity information. This construction of equivalents is closely-related
to the development of vertex sparsifiers, and shares important connections to
several upcoming results in dynamic graph data structures, outside of just the
offline model.
