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When I consider a cycle graph (directed graph), I think the removal of some desired edges is logical. But, If I refer that graph as a plane graph, that mean as a face

then Do the removal of some edges logical with the term face?

Because, I feel the face is a closed entity, so that the term face can not be used when removing edges, in that sense do you think the term cycle is better than face.

What ever the terms face or cycle, my end results are not changing, what is only changing is the explanation.

Please answer me.

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up vote 3 down vote accepted

In a directed planar graph, some faces are surrounded by edges which are cycles. For example:

Planar directed graph with faces shaded

gives rise to the cycles:

Cycle from red face

from the red face (not this is an incoherent cycle; the cycle is weakly connected)

Cycle from green face

from the red face (not this is an coherent cycle; the cycle is strongly connected).

However, I think it's inaccurate to say that "faces" are "cycles" for three reasons:

(1) There are cycles which do not surround faces, e.g. in the following example, the cycle on the right does not surround a face in the graph on the left.

Cycle in a graph that does not surround a face

(2) There are faces in plane graphs which are not surrounded by cycles:

A face which is not surrounded by a cycle

(3) Faces come with some embedding in the plane. Here are two equivalent cycles:

Two isomorphic (i.e., equivalent) cycles

The left-hand side is a cycle that comes from a face. The right-hand side, is just some isomorphic graph drawn differently. They are the same cycle, but would give rise to different faces.

Conclusion: Neither a cycle is a face, nor is a face a cycle.

Other comments:

  • Re: "face is a closed entity", note that the outside face is also considered a face also (and is unbounded).
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