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==Terminology==
If a graph <math>G</math> is embedded on a closed surface <math>\Sigma</math>, the complement of the union of the points and arcs associated with
the vertices and edges of <math>G</math> is a family of '''regions''' (or '''[[face (graph theory)|face]]s''').<ref name="gt01">{{citation|last1=Gross|first1=Jonathan|last2=Tucker|first2=Thomas W.|authorlink2= Thomas W. Tucker| title=Topological Graph Theory|publisher=Dover Publications|year=2001|isbn=0-486-41741-7}}.</ref> A '''2-cell embedding''', '''cellular embedding''' or '''map''' is an embedding in which every face is homeomorphic to an open disk.<ref>{{citation|last1=Lando|first1=Sergei K.|last2=Zvonkin|first2=Alexander K.|title=Graphs on Surfaces and their Applications|publisher=Springer-Verlag|year=2004|isbn=3-540-00203-0}}.</ref> A '''closed 2-cell embedding''' is an embedding in which the closure of every face is homeomorphic to a closed disk.
The '''genus''' of a [[Graph (discrete mathematics)|graph]] is the minimal integer <math>n</math> such that the graph can be embedded in a surface of [[Genus (mathematics)|genus]] <math>n</math>. In particular, a [[planar graph]] has genus <math>0</math>, because it can be drawn on a sphere without self-crossing. The '''non-orientable genus''' of a [[Graph (discrete mathematics)|graph]] is the minimal integer <math>n</math> such that the graph can be embedded in a non-orientable surface of (non-orientable) genus <math>n</math>.<ref name="gt01"/>
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