Face of a plane graph

A face of a plane graph is a connected component of the Euclidean plane after removing the drawn vertices and edges. Count the outer face too. This is the plane counterpart of a face of a planar map on the sphere.^[ref]

The outer face is the unique unbounded face of a plane graph for a finite plane graph. It counts once even though it surrounds the whole graph drawing. A tree has only this face of a plane graph.

For a finite connected graph embedded as a plane graph, follow the edges around a face of a plane graph to obtain a closed walk. Count traversals, not distinct edges: a bridge in a graph occurs twice along one facial boundary walk. Each edge contributes two sides in total.^[ref]

For the three-vertex graph path $A-B-C$, the only facial boundary walk is $A,B,C,B,A$, of length four. Both edges are bridges in a graph; neither is a graph cycle. Thus a facial boundary walk need not be a graph cycle.