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YΔ- And ΔY-transformation
In graph theory, ΔY- and YΔ-transformations (also written delta-wye and wye-delta) are a pair of operations on Graph (discrete mathematics), graphs. A ΔY-transformation replaces a triangle graph, triangle by a vertex of vertex degree, degree three; and conversely, a YΔ-transformation replaces a vertex of degree three by a triangle. The names for the operations derive from the shapes of the involved subgraphs, which look respectively like the letter Y and the Greek capital letter Delta (letter), Δ. A YΔ-transformation may create multi-edge, parallel edges, even if applied to a simple graph. For this reason ΔY- and YΔ-transformations are most naturally considered as operations on multigraphs. On multigraphs both operations preserve the edge count and are exact inverses of each other. In the context of simple graphs it is common to combine a YΔ-transformation with a subsequent ''normalization step'' that reduces parallel edges to a single edge. This may no longer preserve t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forbidden Minor Characterization
In graph theory, a branch of mathematics, many important families of graphs can be described by a finite set of individual graphs that do not belong to the family and further exclude all graphs from the family which contain any of these forbidden graphs as (induced) subgraph or minor. A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar (can be drawn without crossings in the plane) if and only if it does not contain either of two forbidden graphs, the complete graph and the complete bipartite graph . For Kuratowski's theorem, the notion of containment is that of graph homeomorphism, in which a subdivision of one graph appears as a subgraph of the other. Thus, every graph either has a planar drawing (in which case it belongs to the family of planar graphs) or it has a subdivision of at least one of these two graphs as a subgraph (in which case it does not belong to the planar graphs). Definition More generally, a forbidden gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robertson–Seymour Theorem
In graph theory, the Robertson–Seymour theorem (also called the graph minors theorem) states that the undirected graphs, partially ordered by the graph minor relationship, form a well-quasi-ordering. Equivalently, every family of graphs that is closed under taking minors can be defined by a finite set of forbidden minors, in the same way that Wagner's theorem characterizes the planar graphs as being the graphs that do not have the complete graph K_5 or the complete bipartite graph K_ as minors. The Robertson–Seymour theorem is named after mathematicians Neil Robertson and Paul D. Seymour, who proved it in a series of twenty papers spanning over 500 pages from 1983 to 2004. Before its proof, the statement of the theorem was known as Wagner's conjecture after the German mathematician Klaus Wagner, although Wagner said he never conjectured it. A weaker result for trees is implied by Kruskal's tree theorem, which was conjectured in 1937 by Andrew Vázsonyi and proved in 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop (graph Theory)
In graph theory, a loop (also called a self-loop or a ''buckle'') is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a 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. Degree For an undirected graph, the degree of a vertex is equal to the number of adjacent vertices. A special ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apex Rhombic Dodecahedron
The apex is the highest point of something. The word may also refer to: Arts and media Fictional entities * Apex (comics), a teenaged super villainess in the Marvel Universe * Ape-X, a super-intelligent ape in the Squadron Supreme universe *Apex, a genetically engineered human population in the TV series ''The Crossing'' * APEX Medical Hospital, a fictional hospital in the Filipino TV series ''Abot-Kamay na Pangarap'' Music * ''Apex'' (album), by Canadian heavy metal band Unleash the Archers * Apex (band), a Polish heavy metal band * Apex (musician) (1981–2017), British drum and bass music producer and DJ * The Apex Theory, the former name of the alternative rock band Mt. Helium *Lord Apex, a rapper from West London, UK Video games * Apex (tournament), a fighting game tournament focusing on ''Super Smash Bros.'' * '' Racing Evoluzione'', also known as ''Apex'', a 2003 video game for the Xbox * Overwatch Apex, a South Korean ''Overwatch'' tournament series * ''Apex Legends' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heawood Family
In graph theory the term Heawood family refers to either one of the following two related graph families generated via ΔY- and YΔ-transformations: * the family of 20 graphs generated from the complete graph K_7. * the family of 78 graphs generated from K_7 and K_. In either setting the members of the graph family are collectively known as Heawood graphs, as the Heawood graph is a member. This is in analogy to the Petersen family, which too is named after its member the Petersen graph. The Heawood families are significant in topological graph theory. They contain the smallest known examples of intrinsically knotted graphs, of graphs that are not 4-flat, and of graphs with Colin de Verdière graph invariant \mu=6. The K_7-family The K_7-family is generated from the complete graph K_7 through repeated application of ΔY- and YΔ-transformations. The family consists of 20 graphs, all of which have 21 edges. The unique smallest member, K_7, has seven vertices. The unique large ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Graph
In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose. Properties The complete graph on vertices is denoted by . Some sources claim that the letter in this notation stands for the German word , but the German name for a complete graph, , does not contain the letter , and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. has edg ... [...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 YΔ- and ΔY-transformations, 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 YΔ- and ΔY-transformations used to define the Petersen family is as follows: *If a graph contains a vertex with exactly three neighbors, then the YΔ-transform of at is the graph formed by removing from and adding edges between each pair of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
