Editing Homeomorphism (graph theory) (section)

{{Short description|Concept in graph theory}}
{{Distinguish|graph homomorphism}}

In [[graph theory]], two [[graph (discrete mathematics)|graphs]] <math>G</math> and <math>G'</math> are '''homeomorphic''' if there is a [[graph isomorphism]] from some [[#Subdivision_and_smoothing|subdivision]] of <math>G</math> to some subdivision of <math>G'</math>.  If the edges of a graph are thought of as lines drawn from one [[vertex (graph theory)|vertex]] to another (as they are usually depicted in diagrams), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if their diagrams are [[homeomorphism|homeomorphic]] in the [[topology|topological]] sense.<ref>{{citation
 | last = Archdeacon | first = Dan
 | contribution = Topological graph theory: a survey
 | mr = 1411236
 | quote = The name arises because <math>G</math> and <math>H</math> are homeomorphic as graphs if and only if they are homeomorphic as topological spaces
 | pages = 5–54 |citeseerx=10.1.1.28.1728
 | series = Congressus Numerantium
 | title = Surveys in graph theory (San Francisco, CA, 1995)
 | volume = 115
 | year = 1996}}</ref>