The term “plane graph” is used (with a slight bit of ambiguity) to refer to a crossing-free drawing of a graph, or more abstractly to an embedding of a graph into a surface of genus 0 (also called a planar map), or even more abstractly to a graph equipped with a cyclic ordering of the half-edges around each vertex which determines such an embedding up to isomorphism (see: combinatorial map). In any case, when the terminology is employed, it is usually meant to be distinguished from a “planar graph” as being a structure rather than a property (in the sense of stuff, structures, and properties), although sometimes the word “planar graph” is also used in the structure sense.
Sergei K. Lando and Alexander K. Zvonkin, Graphs on Surfaces and Their Applications, Springer, 2004.
Phillipe Flajolet, Robert Sedgewick?: Analytic Combinatorics. First Edition. Cambridge University Press. 2009. (See Chapter VII. 8.2)
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