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.

2 days ago · The genus of a graph is the minimum number of handles that must be added to the plane to embed the graph without any crossings. A graph with genus 0 is embeddable in the plane and is said to be a planar graph. The names of graph classes having particular values for their genera are summarized in the following table (cf. West 2000, p. 266).

We define the genus of a graph in two ways and study the genus of complete graphs and complete bipartite graphs. These topics are also covered in Graph Theory 2

This file contains a moderately-optimized implementation to compute the genus of simple connected graph. It runs about a thousand times faster than the previous version in Sage, not including asymptotic improvements. The algorithm works by enumerating combinatorial embeddings of a graph, and computing the genus of these via the Euler ...

The genus of a graph is the minimal integer such that the graph can be embedded in a surface of genus. In particular, a planar graph has genus 0 {\displaystyle 0} , because it can be drawn on a sphere without self-crossing.

Let $G$ be a finite simple undirected graph. Suppose we contracted some edges of $G$ to form a new graph $G_1$. Then, is it true that the genus of $G$ is greater than or equal to the genus of $G_1$? Thanks in advance.

Every graph has a genus; in fact a graph of size m can be embedded on a surface of genus m. Since the embedding of graphs on spheres and places is equivalent (we can puncture the sphere and get plane), the graphs of genus 0 are precisely the planar graphs.

Every graph has a genus. This result has an easy intuitive verification. Indeed, consider a graph G and any of its plane (or sphere) drawing (possibly with many crossing edges) such that no three edges cross each other in the same point (such a drawing can be obtained).

Oct 27, 2022 · The genus of a graph is a topological invariant that measures the minimum genus of a surface on which the graph can be embedded without any edges crossing. Graph genus plays a fundamental role in topological graph theory, used to classify and study different types of graphs and their properties.

In general, the genus of a torus is the number of holes it has. In fact, the word genus is used for graphs also; the genus of a graph is the minimum number of holes