Worst Case running time of the Minimum Vertex Cover Approximation algorithm
It depends on which data structure you use to store the graph.

For example, suppose the nodes are represented as $0,\ldots,n-1$, and the edges are stored as a list of pairs of nodes. We can build an boolean array $\mathrm{arr}$ of length $n$ initialized with all $\texttt{true}$s, and when a node $i$ is deleted, we simply mark $\mathrm{arr}[i]$ as $\texttt{false}$. Each time we deal with an edge, we first check if any of its endpoints is deleted to see if the edge has already been covered. With such data structures, your algorithm runs in $O(n^2)$.