Given a planar graph (no loops, no multiple edge), is it always possible to perform edge contractions* in order to obtain a graph $T$ which has no loops, and if one ignores parallel edges, $T$ is a tree?
*delete the edge and identify its two ends. This may create loops and multiple edges in the process.
Is there any characterisation of planar graphs which admit such contractions?
I somehow thought there was an easy counterexample, but could not find it. Apologies in advance if this is well-known.