  A maximal independent set (MIS) can be maintained in an evolving $m$-edge
graph by simply recomputing it from scratch in $O(m)$ time after each update.
But can it be maintained in time sublinear in $m$ in fully dynamic graphs?
  We answer this fundamental open question in the affirmative. We present a
deterministic algorithm with amortized update time $O(\min\{\Delta,m^{3/4}\})$,
where $\Delta$ is a fixed bound on the maximum degree in the graph and $m$ is
the (dynamically changing) number of edges.
  We further present a distributed implementation of our algorithm with
$O(\min\{\Delta,m^{3/4}\})$ amortized message complexity, and $O(1)$ amortized
round complexity and adjustment complexity (the number of vertices that change
their output after each update). This strengthens a similar result by
Censor-Hillel, Haramaty, and Karnin (PODC'16) that required an assumption of a
non-adaptive oblivious adversary.
