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Graph-encoded Map
In topological graph theory, a graph-encoded map or gem is a method of encoding a cellular embedding of a graph using a different graph with four vertices per edge of the original graph. It is the topological analogue of runcination, a geometric operation on polyhedra. Graph-encoded maps were formulated and named by . Alternative and equivalent systems for representing cellular embeddings include signed rotation systems and ribbon graphs. The graph-encoded map for an embedded graph G is another cubic graph H together with a 3-edge-coloring of H. Each edge e of G is expanded into exactly four vertices in H, one for each choice of a side and endpoint of the edge. An edge in H connects each such vertex to the vertex representing the opposite side and same endpoint of e; these edges are by convention colored red. Another edge in H connects each vertex to the vertex representing the opposite endpoint and same side of e; these edges are by convention colored blue. An edge in H of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph-encoded Map
In topological graph theory, a graph-encoded map or gem is a method of encoding a cellular embedding of a graph using a different graph with four vertices per edge of the original graph. It is the topological analogue of runcination, a geometric operation on polyhedra. Graph-encoded maps were formulated and named by . Alternative and equivalent systems for representing cellular embeddings include signed rotation systems and ribbon graphs. The graph-encoded map for an embedded graph G is another cubic graph H together with a 3-edge-coloring of H. Each edge e of G is expanded into exactly four vertices in H, one for each choice of a side and endpoint of the edge. An edge in H connects each such vertex to the vertex representing the opposite side and same endpoint of e; these edges are by convention colored red. Another edge in H connects each vertex to the vertex representing the opposite endpoint and same side of e; these edges are by convention colored blue. An edge in H of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Graph
In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. A bicubic graph is a cubic bipartite graph. Symmetry In 1932, Ronald M. Foster began collecting examples of cubic symmetric graphs, forming the start of the Foster census.. Many well-known individual graphs are cubic and symmetric, including the utility graph, the Petersen graph, the Heawood graph, the Möbius–Kantor graph, the Pappus graph, the Desargues graph, the Nauru graph, the Coxeter graph, the Tutte–Coxeter graph, the Dyck graph, the Foster graph and the Biggs–Smith graph. W. T. Tutte classified the symmetric cubic graphs by the smallest integer number ''s'' such that each two oriented paths of length ''s'' can be mapped to each other by exactly one symmetry of the graph. He showed that ''s'' is at most 5, and provided examples of graphs with each possible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Contraction
In graph theory, an edge contraction is an operation that removes an edge from a graph while simultaneously merging the two vertices that it previously joined. Edge contraction is a fundamental operation in the theory of graph minors. Vertex identification is a less restrictive form of this operation. Definition The edge contraction operation occurs relative to a particular edge, e. The edge e is removed and its two incident vertices, u and v, are merged into a new vertex w, where the edges incident to w each correspond to an edge incident to either u or v. More generally, the operation may be performed on a set of edges by contracting each edge (in any order). The resulting induced graph is sometimes written as G/e. (Contrast this with G \setminus e, which means removing the edge e.) As defined below, an edge contraction operation may result in a graph with multiple edges even if the original graph was a simple graph. However, some authors disallow the creation of multip ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface (topology)
In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space. Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis. The various mathematical notions of surface can be used to model surfaces in the physical world. In general In mathematics, a surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries of solid ob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-loop
In graph theory, a loop (also called a self-loop or a ''buckle'') is an edge (graph theory), edge that connects a vertex (graph theory), vertex to itself. A Graph (discrete mathematics)#Simple graph, simple graph contains no loops. Depending on the context, a Graph (discrete mathematics), graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple edges between the same vertices): * Where graphs are defined so as to ''allow'' loops and multiple edges, a graph without loops or multiple edges is often distinguished from other graphs by calling it a ''simple graph''. * Where graphs are defined so as to ''disallow'' loops and multiple edges, a graph that does have loops or multiple edges is often distinguished from the graphs that satisfy these constraints by calling it a ''multigraph'' or ''pseudograph''. In a graph with one vertex, all edges must be loops. Such a graph is called a bouquet graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In 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 conne ..., a tree is an undirected graph in which any two Vertex (graph theory), vertices are connected by ''exactly one'' Path (graph theory), path, or equivalently a Connected graph, connected Cycle (graph theory), 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 graphs, 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 undirecte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flag (geometry)
In (polyhedral) geometry, a flag is a sequence of Face (geometry), faces of a Abstract polytope, polytope, each contained in the next, with exactly one face from each dimension. More formally, a flag of an -polytope is a set such that and there is precisely one in for each , Since, however, the minimal face and the maximal face must be in every flag, they are often omitted from the list of faces, as a shorthand. These latter two are called improper faces. For example, a flag of a polyhedron comprises one Vertex (geometry), vertex, one Edge (geometry), edge incident to that vertex, and one polygonal face incident to both, plus the two improper faces. A polytope may be regarded as regular if, and only if, its symmetry group is transitive group action, transitive on its flags. This definition excludes Chirality (mathematics), chiral polytopes. Incidence geometry In the more abstract setting of incidence geometry, which is a set having a symmetric and reflexive Relatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Coloring
In graph theory, an edge coloring of a graph is an assignment of "colors" to the edges of the graph so that no two incident edges have the same color. For example, the figure to the right shows an edge coloring of a graph by the colors red, blue, and green. Edge colorings are one of several different types of graph coloring. The edge-coloring problem asks whether it is possible to color the edges of a given graph using at most different colors, for a given value of , or with the fewest possible colors. The minimum required number of colors for the edges of a given graph is called the chromatic index of the graph. For example, the edges of the graph in the illustration can be colored by three colors but cannot be colored by two colors, so the graph shown has chromatic index three. By Vizing's theorem, the number of colors needed to edge color a simple graph is either its maximum degree or . For some graphs, such as bipartite graphs and high-degree planar graphs, the number of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ribbon Graph
In topological graph theory, a ribbon graph is a way to represent graph embeddings, equivalent in power to signed rotation systems or graph-encoded maps. It is convenient for visualizations of embeddings, because it can represent unoriented surfaces without self-intersections (unlike embeddings of the whole surface into three-dimensional Euclidean space) and because it omits the parts of the surface that are far away from the graph, allowing holes through which the rest of the embedding can be seen. Ribbon graphs are also called fat graphs. Definition In a ribbon graph representation, each vertex of a graph is represented by a topological disk, and each edge is represented by a topological rectangle with two opposite ends glued to the edges of vertex disks (possibly to the same disk as each other). Embeddings A ribbon graph representation may be obtained from an embedding of a graph onto a surface (and a metric on the surface) by choosing a sufficiently small number \epsilon, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation System
In combinatorial mathematics, rotation systems (also called combinatorial embeddings or combinatorial maps) encode embeddings of graphs onto orientable surfaces by describing the circular ordering of a graph's edges around each vertex. A more formal definition of a rotation system involves pairs of permutations; such a pair is sufficient to determine a multigraph, a surface, and a 2-cell embedding of the multigraph onto the surface. Every rotation scheme defines a unique 2-cell embedding of a connected multigraph on a closed oriented surface (up to orientation-preserving topological equivalence). Conversely, any embedding of a connected multigraph ''G'' on an oriented closed surface defines a unique rotation system having ''G'' as its underlying multigraph. This fundamental equivalence between rotation systems and 2-cell-embeddings was first settled in a dual form by Lothar Heffter in the 1890s and extensively used by Ringel during the 1950s. Independently, Edmonds gave the pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 polyhedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
