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The genus of a graph is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n handles (i.e. an oriented surface of the genus n). Thus, a planar graph has genus 0, because it can be drawn on a sphere without self-crossing. The non-orientable genus of a graph is the minimal integer n such that the graph ...
The Euler genus of a graph is the minimal integer such that the graph can be embedded in an orientable surface of (orientable) genus / or in a non-orientable surface of (non-orientable) genus . A graph is orientably simple if its Euler genus is smaller than its non-orientable genus. The maximum genus of a graph is the maximal integer such that ...
A variable often used to denote a graph. genus The genus of a graph is the minimum genus of a surface onto which it can be embedded; see embedding. geodesic As a noun, a geodesic is a synonym for a shortest path. When used as an adjective, it means related to shortest paths or shortest path distances. giant
More generally, the genus of a graph is the minimum genus of a two-dimensional surface into which the graph may be embedded; planar graphs have genus zero and nonplanar toroidal graphs have genus one. Every graph can be embedded without crossings into some (orientable, connected) closed two-dimensional surface (sphere with handles) and thus the ...
Petersen graph as Kneser graph ,. The Petersen graph is the complement of the line graph of .It is also the Kneser graph,; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an edge if and only if the corresponding 2-element subsets are disjoint from each other.
Bolza surface (genus 2) Klein quartic (genus 3) Bring's curve (genus 4) Macbeath surface (genus 7) Butterfly curve (algebraic) (genus 7) Curve families with variable ...
The hemi-dodecahedron is a regular map produced by pentagonal embedding of the Petersen graph in the projective plane. The p-hosohedron is a regular map of type {2,p}. The Dyck map is a regular map of 12 octagons on a genus-3 surface. Its underlying graph, the Dyck graph, can also form a regular map of 16 hexagons in a torus.
A special kind of spanning tree, the Xuong tree, is used in topological graph theory to find graph embeddings with maximum genus. A Xuong tree is a spanning tree such that, in the remaining graph, the number of connected components with an odd number of edges is as small as possible.