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Robertson–Seymour Theorem
In graph theory, the Robertson–Seymour theorem (also called the graph minor theorem) states that the undirected graphs, partially ordered by the graph minor relationship, form a well-quasi-ordering. Equivalently, every family of graphs that is closed under minors can be defined by a finite set of forbidden minors, in the same way that Wagner's theorem characterizes the planar graphs as being the graphs that do not have the complete graph ''K''5 or the complete bipartite graph ''K''3,3 as minors. The Robertson–Seymour theorem is named after mathematicians Neil Robertson and Paul D. Seymour, who proved it in a series of twenty papers spanning over 500 pages from 1983 to 2004. Before its proof, the statement of the theorem was known as Wagner's conjecture after the German mathematician Klaus Wagner, although Wagner said he never conjectured it. A weaker result for trees is implied by Kruskal's tree theorem, which was conjectured in 1937 by Andrew Vázsonyi and proved in 1960 in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Theory
In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Relation
In mathematics, a binary relation ''R'' on a set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation " is equal to" on the set of real numbers, since every real number is equal to itself. A reflexive relation is said to have the reflexive property or is said to possess reflexivity. Along with symmetry and transitivity, reflexivity is one of three properties defining equivalence relations. Definitions Let R be a binary relation on a set X, which by definition is just a subset of X \times X. For any x, y \in X, the notation x R y means that (x, y) \in R while "not x R y" means that (x, y) \not\in R. The relation R is called if x R x for every x \in X or equivalently, if \operatorname_X \subseteq R where \operatorname_X := \ denotes the identity relation on X. The of R is the union R \cup \operatorname_X, which can equivalently be defined as the smallest (with respect to \subseteq) reflexive relation on X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Path Graph
In the mathematical field of graph theory, a path graph or linear graph is a graph whose vertices can be listed in the order such that the edges are where . Equivalently, a path with at least two vertices is connected and has two terminal vertices (vertices that have degree 1), while all others (if any) have degree 2. Paths are often important in their role as subgraphs of other graphs, in which case they are called paths in that graph. A path is a particularly simple example of a tree, and in fact the paths are exactly the trees in which no vertex has degree 3 or more. A disjoint union of paths is called a linear forest. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See, for example, Bondy and Murty (1976), Gibbons (1985), or Diestel (2005). As Dynkin diagrams In algebra, path graphs appear as the Dynkin diagrams of type A. As such, they classify the root system of type A and the Weyl group of ty ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Union
In mathematics, a disjoint union (or discriminated union) of a family of sets (A_i : i\in I) is a set A, often denoted by \bigsqcup_ A_i, with an injection of each A_i into A, such that the images of these injections form a partition of A (that is, each element of A belongs to exactly one of these images). A disjoint union of a family of pairwise disjoint sets is their union. In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection. In this context, the notation \coprod_ A_i is often used. The disjoint union of two sets A and B is written with infix notation as A \sqcup B. Some authors use the alternative notation A \uplus B or A \operatorname B (along with the corresponding \biguplus_ A_i or \operatorname_ A_i). A standard way for building the disjoint union is to define A as the set of ordered pairs (x, i) such that x \in A_i, and the injection A_i \to A as x \mapsto (x, i). Example Consider the sets A_0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forest (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
