In graph theory, a connected graph $G$ is said to be $3$-vertex-connected (or $3$-connected) if it has more than $3$ vertices and remains connected whenever fewer than $3$ vertices are removed. I was thinking about what the title said above.
problem: Do any two faces of any 3-connected planar graph have at most one edge in common?
My intuitive analysis: Suppose that two faces are associated with at least two edges. If the two edges are adjacent, there exists clearly a 2-cut set.
If not, I think one of the cases is the following:
We still seem to be able to find a 2-cut.
It seems to me that the question I am considering is correct. But my analysis is actually not too strict. I'd like to prove it strictly. Of course it would be nice to find a counterexample.