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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 can be drawn without crossing itself on a sphere with n cross-caps (i.e. a non-orientable surface of (non-orientable) genus n). (This number is also called the demigenus.)
Given a function: from a set X (the domain) to a set Y (the codomain), the graph of the function is the set [4] = {(, ()):}, which is a subset of the Cartesian product.In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph.
A 1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and a k-planar graph is a graph that may be drawn with at most k simple crossings per edge. A map graph is a graph formed from a set of finitely many simply-connected interior-disjoint regions in the plane by connecting two regions when they ...
In algebraic topology and graph theory, graph homology describes the homology groups of a graph, where the graph is considered as a topological space. It formalizes the idea of the number of "holes" in the graph. It is a special case of a simplicial homology, as a graph is a special case of a simplicial complex. Since a finite graph is a 1 ...
Tutte's spring theorem states that every 3-connected planar graph can be drawn on a plane without crossings so that its edges are straight line segments and an outside face is a convex polygon (Tutte 1963). It is so called because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph.
An embedded graph uniquely defines cyclic orders of edges incident to the same vertex. The set of all these cyclic orders is called a rotation system.Embeddings with the same rotation system are considered to be equivalent and the corresponding equivalence class of embeddings is called combinatorial embedding (as opposed to the term topological embedding, which refers to the previous ...
The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a planar graph, the crossing number is zero by definition. Drawings on surfaces other than the plane are also studied.
A general definition of chirality based on group theory exists. [2] It does not refer to any orientation concept: an isometry is direct if and only if it is a product of squares of isometries, and if not, it is an indirect isometry. The resulting chirality definition works in spacetime. [3] [4]