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Factor-critical Graph
In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph.) is a graph with an odd number of vertices in which deleting one vertex in every possible way results in a graph with a perfect matching, a way of grouping the remaining vertices into adjacent pairs. A matching of all but one vertex of a graph is called a near-perfect matching. So equivalently, a factor-critical graph is a graph in which there are near-perfect matchings that avoid every possible vertex. Definition and characterizations Factor-critical graphs may be characterized in several different ways, other than their definition as graphs in which each vertex deletion allows for a perfect matching: * Tibor Gallai proved that a graph is factor-critical if and only if it is connected and, for each node of the graph, there exists a maximum matching that does not include . It follows from these properties that the graph must have an odd number of vertices and that every maximum matchin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factor Critical
Factor (Latin, ) may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, such a factor is a resource used in the production of goods and services * Factor, a brand of HelloFresh meal-kit company Science and technology Biology * Coagulation factors, substances essential for blood coagulation * Environmental factor, any abiotic or biotic factor that affects life * Enzyme, proteins that catalyze chemical reactions * Factor B, and factor D, peptides involved in the alternate pathway of immune system complement activation * Transcription factor, a protein that binds to specific DNA sequences Computer science and information technology * Factor (programming language), a concatenative stack-oriented programming language * Factor (Unix), a utility for factoring an integer into its prime factors * Factor, a substring, a subsequence of consecu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Pentagonal Pyramid
In geometry, the gyroelongated pentagonal pyramid is a polyhedron constructed by attaching a pentagonal antiprism to the base of a pentagonal pyramid. An alternative name is diminished icosahedron because it can be constructed by removing a pentagonal pyramid from a regular icosahedron. Construction The gyroelongated pentagonal pyramid can be constructed from a pentagonal antiprism by attaching a pentagonal pyramid onto its pentagonal face. This pyramid covers the pentagonal faces, so the resulting polyhedron has 15 equilateral triangles and 1 regular pentagon as its faces. Another way to construct it is started from the regular icosahedron by cutting off one of two pentagonal pyramids, a process known as diminishment; for this reason, it is also called the ''diminished icosahedron''. Because the resulting polyhedron has the property of convexity and its faces are regular polygons, the gyroelongated pentagonal pyramid is a Johnson solid, enumerated as the 11th Johnson s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics (journal)
''Discrete Mathematics'' is a biweekly peer-reviewed scientific journal in the broad area of discrete mathematics, combinatorics, graph theory, and their applications. It was established in 1971 and is published by North-Holland Publishing Company. It publishes both short notes, full length contributions, as well as survey articles. In addition, the journal publishes a number of special issues each year dedicated to a particular topic. Although originally it published articles in French and German, it now allows only English language articles. The editor-in-chief is Douglas West ( University of Illinois, Urbana). History The journal was established in 1971. The first article it published was written by Paul Erdős, who went on to publish a total of 84 papers in the journal. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2020 impact factor The impact factor (IF) or journal impact facto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bridge (graph Theory)
In graph theory, a bridge, isthmus, cut-edge, or cut arc is an Glossary of graph theory#edge, edge of a Graph (discrete mathematics), graph whose deletion increases the graph's number of Connected component (graph theory), connected components. Equivalently, an edge is a bridge if and only if it is not contained in any Cycle (graph theory), cycle. For a connected graph, a bridge can uniquely determine a Cut (graph theory), cut. A graph is said to be bridgeless or isthmus-free if it contains no bridges. This type of bridge should be distinguished from an unrelated meaning of "bridge" in graph theory, a subgraph separated from the rest of the graph by a specified subset of vertices; see Glossary of graph theory#bridge, bridge in the Glossary of graph theory. Trees and forests A graph with n nodes can contain at most n-1 bridges, since adding additional edges must create a cycle. The graphs with exactly n-1 bridges are exactly the tree (graph theory), trees, and the graphs in which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 the subgraph G' = (V, E \setminus X) is connected for all X \subseteq E where , X, < k, then ''G'' is said to be ''k''-edge-connected. The edge connectivity of 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 equivalent character ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gyroelongated Pentagonal Pyramid
In geometry, the gyroelongated pentagonal pyramid is a polyhedron constructed by attaching a pentagonal antiprism to the base of a pentagonal pyramid. An alternative name is diminished icosahedron because it can be constructed by removing a pentagonal pyramid from a regular icosahedron. Construction The gyroelongated pentagonal pyramid can be constructed from a pentagonal antiprism by attaching a pentagonal pyramid onto its pentagonal face. This pyramid covers the pentagonal faces, so the resulting polyhedron has 15 equilateral triangles and 1 regular pentagon as its faces. Another way to construct it is started from the regular icosahedron by cutting off one of two pentagonal pyramids, a process known as diminishment; for this reason, it is also called the ''diminished icosahedron''. Because the resulting polyhedron has the property of convexity and its faces are regular polygons, the gyroelongated pentagonal pyramid is a Johnson solid, enumerated as the 11th Johnson s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Pyramid
In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square pyramid'' with four isosceles triangles; otherwise, it is an ''oblique square pyramid''. When all of the pyramid's edges are equal in length, its triangles are all equilateral triangle, equilateral. It is called an ''equilateral square pyramid'', an example of a Johnson solid. Square pyramids have appeared throughout the history of architecture, with examples being Egyptian pyramids and many other similar buildings. They also occur in chemistry in Square pyramidal molecular geometry, square pyramidal molecular structures. Square pyramids are often used in the construction of other polyhedra. Many mathematicians in ancient times discovered the formula for the volume of a square pyramid with different approaches. Special cases Right squar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claw-free Graph
In graph theory, an area of mathematics, a claw-free graph is a graph that does not have a claw (graph theory), claw as an induced subgraph. A claw is another name for the complete bipartite graph K_ (that is, a star graph comprising three edges, three leaves, and a central vertex). A claw-free graph is a graph in which no induced subgraph is a claw; i.e., any subset of four vertices has other than only three edges connecting them in this pattern. Equivalently, a claw-free graph is a graph in which the neighborhood (graph theory), neighborhood of any vertex (graph theory), vertex is the complement (graph theory), complement of a triangle-free graph. Claw-free graphs were initially studied as a generalization of line graphs, and gained additional motivation through three key discoveries about them: the fact that all claw-free connected graphs of even order have perfect matchings, the discovery of polynomial time algorithms for finding maximum independent sets in claw-free graphs, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-vertex-connected Graph
In graph theory, a connected Graph (discrete mathematics), graph is said to be -vertex-connected (or -connected) if it has more than Vertex (graph theory), vertices and remains Connectivity (graph theory), 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. In complete graphs, there is no subset whose removal would disconnect the graph. Some sources modify the definition of connectivity to handle this case, by defining it as the size of the smallest subset of vertices whose deletion results in either a disconnected graph or a single vertex. For this variation, the connectivity of a complete graph K_n is n-1. An equivalent definition is that a graph with at least two vertic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grötzsch Graph
In the mathematical field of graph theory, the Grötzsch graph is a triangle-free graph with 11 vertices, 20 edges, chromatic number 4, and crossing number 5. It is named after German mathematician Herbert Grötzsch, who used it as an example in connection with his 1959 theorem that planar triangle-free graphs are 3-colorable. The Grötzsch graph is a member of an infinite sequence of triangle-free graphs, each the Mycielskian of the previous graph in the sequence, starting from the one-edge graph; this sequence of graphs was constructed by to show that there exist triangle-free graphs with arbitrarily large chromatic number. Therefore, the Grötzsch graph is sometimes also called the Mycielski graph or the Mycielski–Grötzsch graph. Unlike later graphs in this sequence, the Grötzsch graph is the smallest triangle-free graph with its chromatic number. Properties The full automorphism group of the Grötzsch graph is isomorphic to the dihedral group D5 of order 10, the g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mycielskian
In the mathematical area of graph theory, the Mycielskian or Mycielski graph of an undirected graph is a larger graph formed from it by a construction of . The construction preserves the property of being triangle-free but increases the chromatic number; by applying the construction repeatedly to a triangle-free starting graph, Mycielski showed that there exist triangle-free graphs with arbitrarily large chromatic number. Construction Let the ''n'' vertices of the given graph ''G'' be ''v''1, ''v''2, . . . , ''v''n. The Mycielski graph μ(''G'') contains ''G'' itself as a subgraph, together with ''n''+1 additional vertices: a vertex ''u''''i'' corresponding to each vertex ''v''''i'' of ''G'', and an extra vertex ''w''. Each vertex ''u''''i'' is connected by an edge to ''w'', so that these vertices form a subgraph in the form of a star ''K''1,''n''. In addition, for each edge ''v''''i''''v''''j'' of ''G'', the Mycielski graph includes two edges, ''u''''i''''v''''j'' and ''v''''i' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friendship Graph
In the mathematical field of graph theory, the friendship graph (or Dutch windmill graph or -fan) is a planar, undirected graph with vertices and edges. The friendship graph can be constructed by joining copies of the cycle graph with a common vertex, which becomes a universal vertex for the graph. By construction, the friendship graph is isomorphic to the windmill graph . It is unit distance with girth 3, diameter 2 and radius 1. The graph is isomorphic to the butterfly graph. Friendship graphs are generalized by the triangular cactus graphs. Friendship theorem The friendship theorem of states that the finite graphs with the property that every two vertices have exactly one neighbor in common are exactly the friendship graphs. Informally, if a group of people has the property that every pair of people has exactly one friend in common, then there must be one person who is a friend to all the others. However, for infinite graphs, there can be many different graphs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
