Here's a graph problem.

We are given a directed graph. We are allowed to change the directions
of the edges.
Our aim is to minimize the maximum degree in the graph.
How do we achieve this?

One way is to take the vertex with maximum degree, and take another
vertex with least degree reachable from this max-degree vertex and
then reverse all the edges' direction along the path. Now the
questions with this approach are (1) how do we prove that this will
lead to the optimal-graph in the sense, can we get a graph such that
it's maximum degree is the best possible?
(2) What's the time complexity, is it bound tightly?
(3) Is there any better way?

Thanks


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