How to approach Dynamic graph related problems
It's hard to give you concrete techniques because "dynamic" could mean a wide variety of things, and the algorithms/results depend on your model. Below is an overview of the concerns. Here is a paper that gives an overview of some different concerns and models (related to what Peter cited in another answer).



For dynamic problems in general, the key issues are:


do you want an exact solution in all cases, or is approximation allowed?
do you know anything about what changes will occur (e.g. a probability distribution), or are they all equally likely?
how does the algorithm learn about the changes?


A typical dynamic model is something like the following:


Given a graph, you want to compute some property. You are allowed to compute a solution for the initial graph.
You are then told one modification: edge $(e,f)$ is deleted. Compute the property on the new graph using limited resources.
Repeat 2 for $n$ times. 


And here are 3 possible modifications:


You aren't allowed to compute the complete solution at the beginning because of information / time / space limitations (one example of which is online algorithms)
In step 2, the algorithm is not "told" the modification, but has to find the modification in the graph by querying a data structure or something.
A distributed model (like Peter discusses in another answer), where the information is discovered locally and the computation/changes are made locally.


Dynamic models are typically interesting because of resource (e.g. time/space) limitations. In step 2, if I was allowed to compute a complete answer (like I did in step 1) then the problem is easy, since it's just a repeated static graph problem. We're more interested in the smallest amount of resources necessary to compute the change.