Two articulations points connected by an edge. Must this edge be a bridge?

Question

Assume wie have an undirected Graph $G$ (not a multigraph). Assume the vertex $u$ and $v$ are articulation points and there exists an edge $e$ between $u$ and $v$. I believe that the edge $e$ does not have to be a bridge. However, I simply cannot find a counterexample

Answer 1

You are correct. The simplest example is the bull graph; the two vertices of degree three are articulation points, and there is an edge between them.

Answer 2

@Especially Lime has already given the simplest example while I was trying to draw the above graphs in Paint but I still find it insightful to show how to find a counter-example.

The graph $G$ is the first example I could think of where $u$ and $v$ are clearly cut vertices and $e$ is a bridge. Then we can try to add an edge to this graph to make $e$ not a bridge. In order to do that, we can first try to add an edge to the graph (shown in red) where this new edge $f$ connects the components of $G-e$ so that $e$ is not a bridge anymore. Then we can check whether $u$ and $v$ are still cut vertices or not in $G+f$. In my example, they are still cut vertices so we found a counter-example. And if that wasn't the case, we should have added the new edge to somewhere else. In the worst case, we might have to change our first graph $G$.