Dual graph characterization
Task number: 4252
Characterize connected planar graphs that satisfy:
Variant
The dual graph of their arbitrary planar drawing has no loop.
Solution
A loop in a dual graph corresponds to an edge on the boundary of only one face, i.e. to a bridge. If a dual graph does not have a loop, it must be edge-2-connected.
Variant
The dual graph of their arbitrary planar drawing has no loop nor a pair of multiple edges.
Solution
A pair of multiple edges corresponds to two faces that share at least two edges. If a dual graph has neither a loop nor a multiple edge, it must be edge-3-connected.
