K-vertex-connected Graph
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 picture info K-vertex-connected Graph In graph theory, a connected graph is said to be -vertex-connected (or -connected) if it has more than vertices and remains connected whenever fewer than vertices are removed. The vertex-connectivity, or just connectivity, of a graph is the largest for which the graph is -vertex-connected. Definitions A graph (other than a complete graph) has connectivity ''k'' if ''k'' is the size of the smallest subset of vertices such that the graph becomes disconnected if you delete them. Complete graphs are not included in this version of the definition since they cannot be disconnected by deleting vertices. The complete graph with ''n'' vertices has connectivity ''n'' − 1, as implied by the first definition. An equivalent definition is that a graph with at least two vertices is ''k''-connected if, for every pair of its vertices, it is possible to find ''k'' vertex-independent paths connecting these vertices; see Menger's theorem . This definition produces the sam ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info 4-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 , i.e. by 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 ''G'' is therefore disconnected if there exist two vertice ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Steinitz's Theorem In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the 3-vertex-connected planar graphs. That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. This result provides a classification theorem for the three-dimensional convex polyhedra, something that is not known in higher dimensions. It provides a complete and purely combinatorial description of the graphs of these polyhedra, allowing other results on them, such as Eberhard's theorem on the realization of polyhedra with given types of faces, to be proven more easily, without reference to the geometry of these shapes. Additionally, it has been applied in graph drawing, as a way to construct ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Pacific Journal Of Mathematics The Pacific Journal of Mathematics is a mathematics research journal supported by several universities and research institutes, and currently published on their behalf by Mathematical Sciences Publishers, a non-profit academic publishing organisation, and the University of California, Berkeley. It was founded in 1951 by František Wolf and Edwin F. Beckenbach and has been published continuously since, with five two-issue volumes per year and 12 issues per year. Full-text PDF versions of all journal articles are available on-line via the journal's website with a subscription. The journal is incorporated as a 501(c)(3) organization A 501(c)(3) organization is a United States corporation, trust, unincorporated association or other type of organization exempt from federal income tax under section 501(c)(3) of Title 26 of the United States Code. It is one of the 29 types of .... References Mathematics journals Publications established in 1951 Mathematical Sciences Publ ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Vertex Separator In graph theory, a vertex subset is a vertex separator (or vertex cut, separating set) for nonadjacent vertices and if the removal of from the graph separates and into distinct connected components. Examples Consider a grid graph with rows and columns; the total number of vertices is . For instance, in the illustration, , , and . If is odd, there is a single central row, and otherwise there are two rows equally close to the center; similarly, if is odd, there is a single central column, and otherwise there are two columns equally close to the center. Choosing to be any of these central rows or columns, and removing from the graph, partitions the graph into two smaller connected subgraphs and , each of which has at most vertices. If (as in the illustration), then choosing a central column will give a separator with r \leq \sqrt vertices, and similarly if then choosing a central row will give a separator with at most \sqrt vertices. Thus, every grid graph has ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Tutte Embedding In graph drawing and geometric graph theory, a Tutte embedding or barycentric embedding of a simple, 3-vertex-connected, planar graph is a crossing-free straight-line embedding with the properties that the outer face is a convex polygon and that each interior vertex is at the average (or barycenter) of its neighbors' positions. If the outer polygon is fixed, this condition on the interior vertices determines their position uniquely as the solution to a system of linear equations. Solving the equations geometrically produces a planar embedding. Tutte's spring theorem, proven by , states that this unique solution is always crossing-free, and more strongly that every face of the resulting planar embedding is convex. It is called the spring theorem because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph. Example Let ''G'' be the graph of a cube, and (selecting one of its quadrilateral faces as the outer face) ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] Structural Cohesion In sociology, structural cohesion is the conception of a useful formal definition and measure of cohesion in social groups. It is defined as the minimal number of actors in a social network that need to be removed to disconnect the group. It is thus identical to the question of the node connectivity of a given graph in discrete mathematics. The vertex-cut version of Menger's theorem also proves that the disconnection number is equivalent to a maximally sized group with a network in which every pair of persons has at least this number of separate paths between them. It is also useful to know that -cohesive graphs (or -components) are always a subgraph of a -core, although a -core is not always -cohesive. A -core is simply a subgraph in which all nodes have at least neighbors but it need not even be connected. The boundaries of structural endogamy in a kinship group are a special case of structural cohesion. Software Cohesive.blockingis the R program for computing structural ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info K-edge-connected Graph In graph theory, a connected graph is -edge-connected if it remains connected whenever fewer than edges are removed. The edge-connectivity of a graph is the largest for which the graph is -edge-connected. Edge connectivity and the enumeration of -edge-connected graphs was studied by Camille Jordan in 1869. Formal definition Let G = (V, E) be an arbitrary graph. If subgraph G' = (V, E \setminus X) is connected for all X \subseteq E where , X, < k, then ''G'' is ''k''-edge-connected. The edge connectivity of $G$ is the maximum value ''k'' such that ''G'' is ''k''-edge-connected. The smallest set ''X'' whose removal disconnects ''G'' is a minimum cut in ''G''. The edge connectivity version of Menger's theorem provides an alternative and equivale ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Integral Flow Theorem In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem. The maximum value of an s-t flow (i.e., flow from source s to sink t) is equal to the minimum capacity of an s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem. History The maximum flow problem was first formulated in 1954 by T. E. Harris and F. S. Ross as a simplified model of Soviet railway traffic flow. In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford–Fulkerson algorithm.Ford, L.R., Jr.; Fulkerson, D.R., ''Flows in Networks'', Princeton University Press (1962). In their 1955 paper, Ford and Fulkerson wrote that the problem of Harris and Ross is formulated as follows (see p. 5):Consider a rail network con ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Maximum Flow In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen as a special case of more complex network flow problems, such as the circulation problem. The maximum value of an s-t flow (i.e., flow from source s to sink t) is equal to the minimum capacity of an s-t cut (i.e., cut severing s from t) in the network, as stated in the max-flow min-cut theorem. History The maximum flow problem was first formulated in 1954 by T. E. Harris and F. S. Ross as a simplified model of Soviet railway traffic flow. In 1955, Lester R. Ford, Jr. and Delbert R. Fulkerson created the first known algorithm, the Ford–Fulkerson algorithm.Ford, L.R., Jr.; Fulkerson, D.R., ''Flows in Networks'', Princeton University Press (1962). In their 1955 paper, Ford and Fulkerson wrote that the problem of Harris and Ross is formulated as follows (see p. 5):Consider a rail network con ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra. A polyhedron is a 3-dimensional example of a polytope, a more general concept in any number of dimensions. Definition Convex polyhedra are well-defined, with several equivalent standard definitions. However, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of "polyhedron" have been given within particular contexts,. some more rigorous than others, and there is not universal agreement over which of these to choose. Some of these definitions exclude shapes that have often been counted as polyhedra (such as the self-crossing polyhedra) or include shapes that are often not considered as valid polyh ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Planar Graph In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu] picture info Balinski's Theorem In polyhedral combinatorics, a branch of mathematics, Balinski's theorem is a statement about the graph-theoretic structure of three-dimensional convex polyhedra and higher-dimensional convex polytopes. It states that, if one forms an undirected graph from the vertices and edges of a convex ''d''-dimensional convex polyhedron or polytope (its skeleton), then the resulting graph is at least ''d''-vertex-connected: the removal of any ''d'' − 1 vertices leaves a connected subgraph. For instance, for a three-dimensional polyhedron, even if two of its vertices (together with their incident edges) are removed, for any pair of vertices there will still exist a path of vertices and edges connecting the pair. Balinski's theorem is named after mathematician Michel Balinski, who published its proof in 1961, although the three-dimensional case dates back to the earlier part of the 20th century and the discovery of Steinitz's theorem that the graphs of three-dimensional polyh ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]