In a graph, can an edge be in less than 2 faces?

graph-theory

Solution 1:

Consider any tree - it has no cycles, hence its graph has just one (unbounded) face meaning each edge is a part of just a single face. More generally, this is true of any edge which is not part of a cycle.

Solution 2:

Consider the following hastily made graph

            .----.

consisting of $2$ vertices and one edge.

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