How to find largest subgraph G' of graph G in which vertex with lowest degree V exist?

I have a undirected graph. If I have a vertex with lowest degree V there would be 2 subgraphs a subgraph containing v and other subgraph which does not cointain v that is G'=(G-v).

In general I would like to know how to find degree of vertex in undirected graph and a largest subgraph in which vertex with ith degree exist.