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Topological Graph Theory
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. 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 intersecting. A basic embedding problem often presented as a mathematical puzzle is the three utilities problem. Other applications can be found in printing electronic circuits where the aim is to print (embed) a circuit (the graph) on a circuit board (the surface) without two connections crossing each other and resulting in a short circuit. Graphs as topological spaces To an undirected graph we may associate an abstract simplicial complex ''C'' with a single-element set per vertex and a two-element set per edge. The geometric realization , ''C'', of the complex consists of a copy of the unit interval ,1per edge, with the endpoints ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Simplicial Complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. Lee, John M., Introduction to Topological Manifolds, Springer 2011, , p153 For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1). In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems. An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra. Definitions A collection of non-empty finite subsets of a set ''S'' is called a set-family. A set-family is called an abstract simplicial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matching (graph Theory)
In the mathematical discipline of graph theory, a matching or independent edge set in an undirected Graph (discrete mathematics), graph is a set of Edge (graph theory), edges without common vertex (graph theory), vertices. In other words, a subset of the edges is a matching if each vertex appears in at most one edge of that matching. Finding a matching in a bipartite graph can be treated as a Flow network, network flow problem. Definitions Given a Graph (discrete mathematics), graph a matching ''M'' in ''G'' is a set of pairwise non-adjacent edges, none of which are loop (graph theory), loops; that is, no two edges share common vertices. A vertex is matched (or saturated) if it is an endpoint of one of the edges in the matching. Otherwise the vertex is unmatched (or unsaturated). A maximal matching is a matching ''M'' of a graph ''G'' that is not a subset of any other matching. A matching ''M'' of a graph ''G'' is maximal if every edge in ''G'' has a non-empty intersectio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clique (graph Theory)
In graph theory, a clique ( or ) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph G is an induced subgraph of G that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied. Although the study of complete subgraphs goes back at least to the graph-theoretic reformulation of Ramsey theory by , the term ''clique'' comes from , who used complete subgraphs in social networks to model cliques of people; that is, groups of people all of whom know each other. Cliques have many other applications in the sciences and particularly in bioinformatics. Definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Complex
Clique complexes, independence complexes, flag complexes, Whitney complexes and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques (complete subgraphs) of an undirected graph. Clique complex The clique complex of an undirected graph is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the cliques of . Any subset of a clique is itself a clique, so this family of sets meets the requirement of an abstract simplicial complex that every subset of a set in the family should also be in the family. The clique complex can also be viewed as a topological space in which each clique of vertices is represented by a simplex of dimension . The 1-skeleton of (also known as the ''underlying graph'' of the complex) is an undirected graph with a vertex for every 1-element set in the family and an edge for every 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have Group isomorphism, isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface (mathematics), surface), and some point in it, and all the loops both starting and ending at this point—path (topology), paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then alo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,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 arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a Subspace topology, subspace of X. Some related but stronger conditions are #Path connectedness, path connected, Simply connected space, simply connected, and N-connected space, n-connected. Another related notion is Locally connected space, locally connected, which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Graph
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. Connected vertices and graphs In an undirected graph , two vertices and are called connected if contains a path from to . Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length (that is, they are the endpoints of a single edge), the vertices are called adjacent. A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected. An undirected graph is therefore disconnected if there e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphism (graph Theory)
In graph theory, two graph (discrete mathematics), graphs G and G' are homeomorphic if there is a graph isomorphism from some #Subdivision_and_smoothing, subdivision of G to some subdivision of G'. If the edges of a graph are thought of as lines drawn from one vertex (graph theory), vertex to another (as they are usually depicted in diagrams), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if their diagrams are homeomorphism, homeomorphic in the topology, topological sense. Subdivision and smoothing In general, a subdivision of a graph ''G'' (sometimes known as an expansion) is a graph resulting from the subdivision of edges in ''G''. The subdivision of some edge ''e'' with endpoints yields a graph containing one new vertex ''w'', and with an edge set replacing ''e'' by two new edges, and . For directed edges, this operation shall preserve their propagating direction. For example, the edge ''e'', with endpoints : can be subdivided int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
