Editing Planar graph (section)

===Outerplanar graphs===
[[Outerplanar graph]]s are graphs with an embedding in the plane such that all vertices belong to the unbounded face of the embedding. Every outerplanar graph is planar, but the converse is not true: {{math|''K''<sub>4</sub>}} is planar but not outerplanar. A theorem similar to Kuratowski's states that a finite graph is outerplanar if and only if it does not contain a subdivision of {{math|''K''<sub>4</sub>}} or of {{math|''K''<sub>2,3</sub>}}. The above is a direct corollary of the fact that a graph {{mvar|G}} is outerplanar if the graph formed from {{mvar|G}} by adding a new vertex, with edges connecting it to all the other vertices, is a planar graph.<ref>{{citation
 | last = Felsner | first = Stefan
 | contribution = 1.4 Outerplanar Graphs and Convex Geometric Graphs
 | doi = 10.1007/978-3-322-80303-0_1
 | isbn = 3-528-06972-4
 | mr = 2061507
 | pages = 6–7
 | publisher = Friedr. Vieweg & Sohn, Wiesbaden
 | series = Advanced Lectures in Mathematics
 | title = Geometric graphs and arrangements
 | year = 2004}}</ref>

A 1-outerplanar embedding of a graph is the same as an outerplanar embedding.  For {{math|''k'' > 1}} a planar embedding is {{mvar|k}}-outerplanar if removing the vertices on the outer face results in a {{math|(''k'' − 1)}}-outerplanar embedding.  A graph is {{mvar|k}}-outerplanar if it has a {{mvar|k}}-outerplanar embedding.