Greedy algorithm for vertex cover Given a graph $G(V, E)$, consider the following algorithm:

Let $d$ be the minimum vertex degree of the graph (ignore vertices with degree 0, so that $d\geq 1$)
Let $v$ be one of the vertices with degree equal to $d$
Remove all vertices adjacent to $v$ and add them to the proposed vertex cover
Repeat from step 1. until in $G$ there are only vertices with degree $0$ (no edges in the graph)

At the end the removed vertices are a vertex cover of the given $G(V, E)$, but is it a minimum vertex cover? Is there an example where the algorithm does not find a minimum vertex cover?
Start with a clique on the vertices $A,B,C,D$.
Connect $A,B$ to a new vertex $a$.
Connect $B,C$ to a new vertex $c$.
Connect $a,c$ to a new vertex $b$.
The vertex $b$ is the only one of degree 2, so your algorithm will start by adding $a,c$ to the vertex cover. The remaining graph consists of the clique $A,B,C,D$ and the isolated vertex $b$, so the algorithm will add 3 more vertices, ending up with a vertex cover of size 5.
In contrast, the 4 vertices $A,B,C,b$ cover all edges.