  We introduce a new algorithmic framework for designing dynamic graph
algorithms in minor-free graphs, by exploiting the structure of such graphs and
a tool called vertex sparsification, which is a way to compress large graphs
into small ones that well preserve relevant properties among a subset of
vertices and has previously mainly been used in the design of approximation
algorithms.
  Using this framework, we obtain a Monte Carlo randomized fully dynamic
algorithm for $(1+\varepsilon)$-approximating the energy of electrical flows in
$n$-vertex planar graphs with $\tilde{O}(r\varepsilon^{-2})$ worst-case update
time and $\tilde{O}((r+\frac{n}{\sqrt{r}})\varepsilon^{-2})$ worst-case query
time, for any $r$ larger than some constant. For $r=n^{2/3}$, this gives
$\tilde{O}(n^{2/3}\varepsilon^{-2})$ update time and
$\tilde{O}(n^{2/3}\varepsilon^{-2})$ query time. We also extend this algorithm
to work for minor-free graphs with similar approximation and running time
guarantees. Furthermore, we illustrate our framework on the all-pairs max flow
and shortest path problems by giving corresponding dynamic algorithms in
minor-free graphs with both sublinear update and query times. To the best of
our knowledge, our results are the first to systematically establish such a
connection between dynamic graph algorithms and vertex sparsification.
  We also present both upper bound and lower bound for maintaining the energy
of electrical flows in the incremental subgraph model, where updates consist of
only vertex activations, which might be of independent interest.
