  We consider the problem of dynamically maintaining (approximate) all-pairs
effective resistances in separable graphs, which are those that admit an
$n^{c}$-separator theorem for some $c<1$. We give a fully dynamic algorithm
that maintains $(1+\varepsilon)$-approximations of the all-pairs effective
resistances of an $n$-vertex graph $G$ undergoing edge insertions and deletions
with $\tilde{O}(\sqrt{n}/\varepsilon^2)$ worst-case update time and
$\tilde{O}(\sqrt{n}/\varepsilon^2)$ worst-case query time, if $G$ is guaranteed
to be $\sqrt{n}$-separable (i.e., it is taken from a class satisfying a
$\sqrt{n}$-separator theorem) and its separator can be computed in
$\tilde{O}(n)$ time. Our algorithm is built upon a dynamic algorithm for
maintaining \emph{approximate Schur complement} that approximately preserves
pairwise effective resistances among a set of terminals for separable graphs,
which might be of independent interest.
  We complement our result by proving that for any two fixed vertices $s$ and
$t$, no incremental or decremental algorithm can maintain the $s-t$ effective
resistance for $\sqrt{n}$-separable graphs with worst-case update time
$O(n^{1/2-\delta})$ and query time $O(n^{1-\delta})$ for any $\delta>0$, unless
the Online Matrix Vector Multiplication (OMv) conjecture is false.
  We further show that for \emph{general} graphs, no incremental or decremental
algorithm can maintain the $s-t$ effective resistance problem with worst-case
update time $O(n^{1-\delta})$ and query-time $O(n^{2-\delta})$ for any $\delta
>0$, unless the OMv conjecture is false.
