  Most graphs in real life keep changing with time. These changes can be in the
form of insertion or deletion of edges or vertices. Such rapidly changing
graphs motivate us to study dynamic graph algorithms. However, three important
graph problems that are perhaps not sufficiently addressed in the literature
include independent sets, maximum matching (exact) and maximum flows.
  Maximal Independent Set (MIS) is one of the most prominently studied problems
in the distributed setting. Recently, the first dynamic MIS algorithm for
distributed networks was given by Censor-Hillel et al. [PODC16], requiring
expected $O(1)$ amortized rounds with $O(\Delta)$ messages per update, where
$\Delta$ is the maximum degree of a vertex in the graph. They suggested an open
problem to maintain MIS in fully dynamic centralized setting more efficiently.
Assadi et al. [STOC18] presented a deterministic centralized fully dynamic MIS
algorithm requiring $O(\min\{\Delta,m^{3/4}\})$ amortized time per update. This
result is quite complex involving an exhaustive case analysis. We report a
surprisingly simple deterministic centralized algorithm which improves the
amortized update time to $O(\min\{\Delta,m^{2/3}\})$.
  Additionally, we present some other minor results related to dynamic MIS,
Maximum Flow, and Maximum Matching. A common trait of all our results is that
despite improving state of the art upper bounds or matching state of the art
lower bounds, they are surprisingly simple and are analysed using simple
amortization arguments. Further, they use no complicated data structures or
black box algorithms for their implementation.
