Editing Planar graph (section)

{{Redirect|Triangular graph|line graphs of complete graphs|Line graph#Strongly regular and perfect line graphs|triangulated graphs|Chordal graph|data graphs plotted across three variables|Ternary plot}}
{{short description|Graph that can be embedded in the plane}}
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{|class="wikitable" align="right" style="margin-left: 1em;"
!colspan="2"|Example graphs
|-
! Planar
! Nonplanar
|-
| align="center" | [[Image:Butterfly graph.svg|100px]] <br> [[Butterfly graph]]
| align="center" | [[Image:Complete graph K5.svg|100px]] <br>[[Complete graph]] ''K''<sub>5</sub>
|-
| align="center" | [[File:CGK4PLN.svg|100x100px]] <br> [[Complete graph]]<br> ''K''<sub>4</sub> 
| align="center" | [[Image:Biclique K 3 3.svg|100px]] <br>[[Utility graph]] ''K''<sub>3,3</sub>
|}

In [[graph theory]], a '''planar graph''' is a [[graph (discrete mathematics)|graph]] that can be [[graph embedding|embedded]] in the [[plane (geometry)|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.<ref>{{cite book|last=Trudeau|first=Richard J.|title=Introduction to Graph Theory|year=1993|publisher=Dover Pub.|location=New York|isbn=978-0-486-67870-2|pages=64|url=http://store.doverpublications.com/0486678709.html|edition=Corrected, enlarged republication.|access-date=8 August 2012|quote=Thus a planar graph, when drawn on a flat surface, either has no edge-crossings or can be redrawn without them.}}</ref><ref>{{cite book |last1=Barthelemy |first1=M. |chapter=1.5 Planar Graphs |chapter-url={{GBurl|9-hEDwAAQBAJ|p=6}}  |title=Morphogenesis of Spatial Networks |date=2017 |isbn=978-3-319-20565-6 |publisher=Springer |page=6}}</ref>   Such a drawing is called a '''plane graph''', or a '''planar embedding''' of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a [[plane curve]] on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

Every graph that can be drawn on a plane can be drawn on the [[sphere]] as well, and vice versa, by means of [[stereographic projection]].

Plane graphs can be encoded by [[combinatorial map]]s or [[rotation system]]s.

An [[equivalence class]] of [[topologically equivalent]] drawings on the sphere, usually with additional assumptions such as the absence of [[bridge (graph theory)|isthmus]]es, is called a '''planar map'''. Although a plane graph has an '''external''' or '''unbounded''' [[face (graph theory)|face]], none of the faces of a planar map has a particular status.

Planar graphs generalize to graphs drawable on a surface of a given [[genus (mathematics)|genus]]. In this terminology, planar graphs have [[graph genus|genus]]&nbsp;0, since the plane (and the sphere) are surfaces of genus&nbsp;0. See "[[graph embedding]]" for other related topics.