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I am bit confused with graph theory terms. Could you please tell me, if I say

  1. plane embedded graph
  2. embedded plane graph

what is correct.

Also, If i use either 1 or 2 (the correct one) to refer a planar graph, then does it correct?

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up vote 1 down vote accepted

When it comes to mathematics, we do our best to write unambiguously, but it's not always possible to do so while being able to communicate ideas well. At some point we need to say enough is enough, and an intelligent reader will be able to find the correct interpretation themselves from what we've presented.

To illustrate, consider what happens if we have a multigraph and delete the parallel edges. The usual interpretation of this is not the literal one. Now imagine what would happen if we attempted to re-write this without the ambiguity: we'd make it technically correct, but much more difficult to understand.

I'd say you're in this situation. Both plane embedded graph and embedded plane graph are pretty close, and an intelligent reader (who's not dead set on nit-picking) should be able to understand what you mean. It's somewhat tautological, since plane graph implies it comes with an embedding in the plane, so plane graph would probably be better.

A plane graph distinguishes itself from a planar graph in that it has a given embedding. E.g. $K_4$ is a planar graph, but $K_4$ by itself is not a plane graph.

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"In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph."

That is from Wikipedia. I hope that sheds light on how to properly use these terms and refer to these properties.

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