I want to determine if a given graph is a minimal 3-connected graph. That is, deletion of any edge will reduce the vertex connectivity to 2.

My approach right now, is to look at every edge where both endpoints have degree 4 or more, remove the edge and see if the vertex connectivity has decreased to 2 in the whole graph.

My question is, can I use the following approach instead: Look at every edge $e=xy$ where both endpoints have degree $\geq$ 4. Remove $e$ and check if the connectivity between $x$ and $y$ has decreased.

So I want to know if I can reduce the problem to considering only the connectivity between the endpoints of the edge I am removing.