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In mathematics, an embedding (or imbedding [1]) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X {\displaystyle X} is said to be embedded in another object Y {\displaystyle Y} , the embedding is given by some injective and structure-preserving map f : X → ...
In mathematics, topological graph theory is a branch of graph theory. It studies the embedding of graphs in surfaces, spatial embeddings of graphs, and graphs as topological spaces. [1] It also studies immersions of graphs. Embedding a graph in a surface means that we want to draw the graph on a surface, a sphere for example, without two edges ...
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 surface onto which the graph is embedded may be determined by whether it is orientable (true if any cycle in the graph has an even number of twists) and by its Euler characteristic. The embeddings that can be represented by ribbon graphs are the ones in which a graph is embedded onto a 2- manifold (without boundary) and in which each face ...
A smooth embedding is an injective immersion f : M → N that is also a topological embedding, so that M is diffeomorphic to its image in N. An immersion is precisely a local embedding – that is, for any point x ∈ M there is a neighbourhood, U ⊆ M, of x such that f : U → N is an embedding, and conversely a local embedding is an ...
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: . The strong Whitney embedding theorem states that any smooth real m-dimensional manifold (required also to be Hausdorff and second-countable) can be smoothly embedded in the real 2m-space, , if m > 0.
A toroidal graph is a graph that can be embedded without crossings on the torus. 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 ...
The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.