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Petersen Family
In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph . The Petersen family is named after Danish mathematician Julius Petersen, the namesake of the Petersen graph. Any of the graphs in the Petersen family can be transformed into any other graph in the family by Δ-Y or Y-Δ transforms, operations in which a triangle is replaced by a degree-three vertex or vice versa. These seven graphs form the forbidden minors for linklessly embeddable graphs, graphs that can be embedded into three-dimensional space in such a way that no two cycles in the graph are linked. They are also among the forbidden minors for the YΔY-reducible graphs. Definition The form of Δ-Y and Y-Δ transforms used to define the Petersen family is as follows: *If a graph contains a vertex with exactly three neighbors, then the Y-Δ transform of at is the graph formed by removing from and adding edges between each pair of its thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersen Family
In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph . The Petersen family is named after Danish mathematician Julius Petersen, the namesake of the Petersen graph. Any of the graphs in the Petersen family can be transformed into any other graph in the family by Δ-Y or Y-Δ transforms, operations in which a triangle is replaced by a degree-three vertex or vice versa. These seven graphs form the forbidden minors for linklessly embeddable graphs, graphs that can be embedded into three-dimensional space in such a way that no two cycles in the graph are linked. They are also among the forbidden minors for the YΔY-reducible graphs. Definition The form of Δ-Y and Y-Δ transforms used to define the Petersen family is as follows: *If a graph contains a vertex with exactly three neighbors, then the Y-Δ transform of at is the graph formed by removing from and adding edges between each pair of its thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minor (graph Theory)
Minor may refer to: * Minor (law), a person under the age of certain legal activities. ** A person who has not reached the age of majority * Academic minor, a secondary field of study in undergraduate education Music theory *Minor chord ** Barbershop seventh chord or minor seventh chord *Minor interval *Minor key *Minor scale Mathematics * Minor (graph theory), the relation of one graph to another given certain conditions * Minor (linear algebra), the determinant of a certain submatrix People * Charles Minor (1835–1903), American college administrator * Charles A. Minor (21st-century), Liberian diplomat * Dan Minor (1909–1982), American jazz trombonist * Dave Minor (1922–1998), American basketball player * James T. Minor, US academic administrator and sociologist * Jerry Minor (born 1969), American actor, comedian and writer * Kyle Minor (born 1976), American writer * Mike Minor (actor) (born 1940), American actor * Mike Minor (baseball) (born 1987), American baseball p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apex Graph
In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is ''an'' apex, not ''the'' apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs or , every vertex is an apex. The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove. Apex graphs are closed under the operation of taking minors and play a role in several other aspects of graph minor theory: linkless embedding, Hadwiger's conjecture,. YΔY-reducible graphs, and relations between treewidth and graph diameter. Characterization and recognition Apex graphs are closed under the operation of taking minors: contracting any edge, or removing any edge or vertex, leads to another apex graph. For, if is an apex graph with apex , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star (graph Theory)
In graph theory, a star is the complete bipartite graph a tree with one internal node and leaves (but no internal nodes and leaves when ). Alternatively, some authors define to be the tree of order with maximum diameter 2; in which case a star of has leaves. A star with 3 edges is called a claw. The star is edge-graceful when is even and not when is odd. It is an edge-transitive matchstick graph, and has diameter 2 (when ), girth ∞ (it has no cycles), chromatic index , and chromatic number 2 (when ). Additionally, the star has large automorphism group, namely, the symmetric group on letters. Stars may also be described as the only connected graphs in which at most one vertex has degree greater than one. Relation to other graph families Claws are notable in the definition of claw-free graphs, graphs that do not have any claw as an induced subgraph. They are also one of the exceptional cases of the Whitney graph isomorphism theorem: in general, graphs with isomor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horst Sachs
Horst Sachs (27 March 1927 – 25 April 2016) was a German mathematician, an expert in graph theory, a recipient of the Euler Medal (2000). He earned the degree of Doctor of Science (Dr. rer. nat.) from the Martin-Luther-Universität Halle-Wittenberg in 1958. Following his retirement in 1992, he was professor emeritus at the Institute of Mathematics of the Technische Universität Ilmenau. His encyclopedic book in spectral graph theory, ''Spectra of Graphs. Theory and Applications'' (with Dragos Cvetković and Michael Doob) has several editions and was translated in several languages.Review by P. Rowlinson (1996), ''Proceedings of the Edinburgh Mathematical Society (Series 2)'' 39: 188–189, . Two theorems in graph theory bear his name. One of them relates the coefficients of the characteristic polynomial of a graph to certain structural features of the graph. Another one is a simple relation between the characteristic polynomials of a graph and its line graph In the math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a ''directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed graph. A directed circuit is a non-empty directed trail with a vertex sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the Greek mathematics, ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of ''mathematical proof, proving'' all properties of the space as theorems, by starting from a few fundamental properties, called ''postulates'', which either were considered as eviden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robin Thomas (mathematician)
Robin Thomas (August 22, 1962 – March 26, 2020) was a mathematician working in graph theory at the Georgia Institute of Technology. Thomas received his doctorate in 1985 from Charles University in Prague, Czechoslovakia (now the Czech Republic), under the supervision of Jaroslav Nešetřil. He joined the faculty at Georgia Tech in 1989, and became a Regents' Professor there, briefly serving as the department Chair. On March 26, 2020, he died of Amyotrophic Lateral Sclerosis at the age of 57 after 12 years of struggle with the illness. Awards Thomas was awarded the Fulkerson Prize for outstanding papers in discrete mathematics twice, in 1994 as co-author of a paper on the Hadwiger conjecture, and in 2009 for the proof of the strong perfect graph theorem. In 2011 he was awarded the Karel Janeček Foundation Neuron Prize for Lifetime Achievement in Mathematics. In 2012 he became a fellow of the American Mathematical Society. He was named a SIAM Fellow The SIAM Fellowship is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Seymour (mathematician)
Paul D. Seymour (born 26 July 1950) is a British mathematician known for his work in discrete mathematics, especially graph theory. He (with others) was responsible for important progress on regular matroids and totally unimodular matrices, the four colour theorem, linkless embeddings, graph minors and structure, the perfect graph conjecture, the Hadwiger conjecture, claw-free graphs, χ-boundedness, and the Erdős–Hajnal conjecture. Many of his recent papers are available from his website. Seymour is currently the Albert Baldwin Dod Professor of Mathematics at Princeton University. He won a Sloan Fellowship in 1983, and the Ostrowski Prize in 2004; and (sometimes with others) won the Fulkerson Prize in 1979, 1994, 2006 and 2009, and the Pólya Prize in 1983 and 2004. He received an honorary doctorate from the University of Waterloo in 2008, one from the Technical University of Denmark in 2013, and one from the École normale supérieure de Lyon in 2022. He was an invited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neil Robertson (mathematician)
George Neil Robertson (born November 30, 1938) is a mathematician working mainly in topological graph theory, currently a distinguished professor emeritus at the Ohio State University. Education Robertson earned his B.Sc. from Brandon College in 1959, and his Ph.D. in 1969 at the University of Waterloo under his doctoral advisor William Tutte. Biography In 1969, Robertson joined the faculty of the Ohio State University, where he was promoted to Associate Professor in 1972 and Professor in 1984. He was a consultant with Bell Communications Research from 1984 to 1996. He has held visiting faculty positions in many institutions, most extensively at Princeton University from 1996 to 2001, and at Victoria University of Wellington, New Zealand, in 2002. He also holds an adjunct position at King Abdulaziz University in Saudi Arabia.. Research Robertson is known for his work in graph theory, and particularly for a long series of papers co-authored with Paul Seymour and published over a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wagner's Theorem
In graph theory, Wagner's theorem is a mathematical forbidden graph characterization of planar graphs, named after Klaus Wagner, stating that a finite graph is planar if and only if its minors include neither ''K''5 (the complete graph on five vertices) nor ''K''3,3 (the utility graph, a complete bipartite graph on six vertices). This was one of the earliest results in the theory of graph minors and can be seen as a forerunner of the Robertson–Seymour theorem. Definitions and statement A planar embedding of a given graph is a drawing of the graph in the Euclidean plane, with points for its vertices and curves for its edges, in such a way that the only intersections between pairs of edges are at a common endpoint of the two edges. A minor of a given graph is another graph formed by deleting vertices, deleting edges, and contracting edges. When an edge is contracted, its two endpoints are merged to form a single vertex. In some versions of graph minor theory the graph r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
