A crossing is a pair of intersecting edges. A graph is embeddable on a surface if its vertices and edges can be arranged on it without any crossing. The genus of a graph is the lowest genus of any surface on which the graph can embed. A planar graph is one which can be drawn on the (Euclidean) plane without any crossing; and a plane graph, one which is drawn in such fashion. In other words, a planar graph is a graph of genus 0. The example graph is planar; the complete graph on n vertices, for n> 4, is not p ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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An arc, or directed edge, is an ordered pair of endvertices. In such ordered pair, the first vertex is called a head, or initial vertex; and the second one, a tail, or terminal vertex. It can be thought of as an edge associated with a direction, namely designating a head and a tail to the endvertices. An undirected edge disregards any sense of direction and treats both endvertices interchangeably. A loop in a digraph, however, keeps a sense of direction and treats both head and tail ide ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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The distance dG(u, v) between two (not necessary distinct) vertices u and v in a graph G is the length of a shortest path between them. The subscript G is usually dropped when there is no danger of confusion. When u and v are identical, their distance is 0. When u and v are unreachable from each ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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Connectivity extends the concept of adjacency and is essentially a form (and measure) of concatenated adjacency. If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is said to be connected; otherwise, the graph is disconnected. A graph is totally disconnected if there is no path connecting any pair of vertices. This is just an ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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Sometimes pattern is defined as the repetition of a template, or as the abstract quality shared by products produced from a template according to a set of rules. In sewing and fashion design, a pattern is: An original article of clothing from which others are copied The (usually) paper template used to cut out the parts of a garment before assembling (see Pattern (sewing)) In machine learning, a pattern is a non-null finite sequence of constant and variable symbols. The language of a pattern is the set of strings generated by substituting variables in the ...

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Pattern, Pattern - Patterns in nature, Pattern - Patterns in art, Pattern - Patterns in science and mathematics, Pattern - Quotation, Pattern - Other senses of pattern

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In graph theory, degree, especially that of a vertex, is usually a measure of immediate adjacency. An edge connects two vertices; these two vertices are said to be incident to that edge, or, equivalently, that edge incident to those two vertices. All degree-related concepts have to do with adjacency or incidence. The degree, or valency, dG(v) of a vertex v in a graph G is the number of edges incident to v, with loops being counted twice. A vertex of degre ...

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Glossary of graph theory, Glossary of graph theory - Basics, Glossary of graph theory - Subgraphs, Glossary of graph theory - Paths and walks, Glossary of graph theory - Cycles, Glossary of graph theory - Trees, Glossary of graph theory - Cliques, Glossary of graph theory - Strongly connected component, Glossary of graph theory - Knots, Glossary of graph theory - Minors, Glossary of graph theory - Embedding, Glossary of graph theory - Adjacency and degree, Glossary of graph theory - Independence, Glossary of graph theory - Connectivity, Glossary of graph theory - Distance, Glossary of graph theory - Genus, Glossary of graph theory - Weighted graphs and networks, Glossary of graph theory - Direction, Glossary of graph theory - Various, Glossary of graph theory - To be merged

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