The dual graph of a planar graph is the graph formed by placing one vertex in every cell and one edge between the vertices of adjacent cells.
A graph is 2-vertex-connected if removing any one vertex does not disconnect the graph, and 3-vertex connected if removing any two vertices does not disconnect it. It is 3-regular if every vertex has degree 3.
Let G be a graph that is planar, 2-vertex connected, and 3-regular, but not 3-vertex-connected.
Does the dual of G have at least one pair of vertices with two or more edges between them? In other words, does the dual of G necessarily have a multiedge?
*Edit: neglected to mention the lack of 3-vertex-connectivity.