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Planar Graphs
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 a 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 calle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butterfly Graph
In the mathematics, mathematical field of graph theory, the butterfly graph (also called the bowtie graph and the hourglass graph) is a planar graph, planar, undirected graph with 5 Vertex (graph theory), vertices and 6 edges. It can be constructed by joining 2 copies of the cycle graph with a common vertex and is therefore Graph isomorphism, isomorphic to the friendship graph . The butterfly graph has graph diameter, diameter 2 and girth (graph theory), girth 3, radius 1, chromatic number 3, chromatic index 4 and is both Eulerian graph, Eulerian and a penny graph (this implies that it is unit distance graph, unit distance and planar graph, planar). It is also a 1-k-vertex-connected graph, vertex-connected graph and a 2-k-edge-connected graph, edge-connected graph. There are only three Graceful labeling, non-graceful simple graphs with five vertices. One of them is the butterfly graph. The two others are cycle graph and the complete graph . Bowtie-free grap ... [...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] |
Complete Bipartite Graph
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set..Electronic edition page 17. Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete bipartite graphs were already printed as early as 1669, in connection with an edition of the works of Ramon Llull edited by Athanasius Kircher. Llull himself had made similar drawings of complete graphs three centuries earlier.. Definition A complete bipartite graph is a graph whose vertices can be partitioned into two subsets and such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. That is, it is a bipartite graph such that for every two vertices and, is an edge in . A complete bipartite graph w ... [...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] |
Subdivision (graph Theory)
In graph theory, two graphs G and G' are homeomorphic if there is a graph isomorphism from some subdivision of G to some subdivision of G'. If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in diagrams), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if their diagrams are homeomorphic in the topological sense. Subdivision and smoothing In general, a subdivision of a graph ''G'' (sometimes known as an expansion) is a graph resulting from the subdivision of edges in ''G''. The subdivision of some edge ''e'' with endpoints yields a graph containing one new vertex ''w'', and with an edge set replacing ''e'' by two new edges, and . For directed edges, this operation shall preserve their propagating direction. For example, the edge ''e'', with endpoints : can be subdivided into two edges, ''e''1 and ''e''2, connecting to a new vertex ''w'' of degree-2, or indegree-1 and o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary Of Graph Theory
This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges. Symbols A B C D E F G H I J K L M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
If And Only If
In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false. The connective is biconditional (a statement of material equivalence), and can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false), though it is controversial whether the connective thus defined is properly rendered by the English "if and only if"—with its pre-existing meaning. For example, ''P if and only if Q'' means that ''P'' is true whenever ''Q'' is true, and the only case in which ''P'' is true is if ''Q'' is also true, whereas in the case of ''P if Q ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Graph
In discrete mathematics, particularly in graph theory, a graph is a structure consisting of a set of objects where some pairs of the objects are in some sense "related". The objects are represented by abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', then this graph is directed, because owing money is not necessarily reciprocated. Gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuratowski's Theorem
In graph theory, Kuratowski's theorem is a mathematical forbidden graph characterization of planar graphs, named after Kazimierz Kuratowski. It states that a finite graph is planar if and only if it does not contain a Glossary of graph theory#Subgraphs, subgraph that is a subdivision (graph theory), subdivision of K_5 (the complete graph on five vertex (graph theory), vertices) or of K_ (a complete bipartite graph on six vertices, three of which connect to each of the other three, also known as the utility graph). Statement A planar graph is a graph whose vertices can be represented by points in the Euclidean plane, and whose edges can be represented by simple curves in the same plane connecting the points representing their endpoints, such that no two curves intersect except at a common endpoint. Planar graphs are often graph drawing, drawn with straight line segments representing their edges, but by Fáry's theorem this makes no difference to their graph-theoretic characterizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forbidden Graph Characterization
In graph theory, a branch of mathematics, many important families of Graph (discrete mathematics), 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) Glossary_of_graph_theory#subgraph, subgraph or Graph minor, minor. A prototypical example of this phenomenon is Kuratowski's theorem, which states that a graph is planar graph, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kazimierz Kuratowski
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. He worked as a professor at the University of Warsaw and at the Mathematical Institute of the Polish Academy of Sciences (IM PAN). Between 1946 and 1953, he served as President of the Polish Mathematical Society. He is primarily known for his contributions to set theory, topology, measure theory and graph theory. Some of the notable mathematical concepts bearing Kuratowski's name include Kuratowski's theorem, Kuratowski closure axioms, Kuratowski-Zorn lemma and Kuratowski's intersection theorem. Life and career Early life Kazimierz Kuratowski was born in Warsaw, (then part of Congress Poland controlled by the Russian Empire), on 2 February 1896. He was a son of Marek Kuratow, a barrister, and Róża Karzewska. He completed a Warsaw secondary school, which was named after general Paweł Chrzanowski. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poland
Poland, officially the Republic of Poland, is a country in Central Europe. It extends from the Baltic Sea in the north to the Sudetes and Carpathian Mountains in the south, bordered by Lithuania and Russia to the northeast, Belarus and Ukraine to the east, Slovakia and the Czech Republic to the south, and Germany to the west. The territory has a varied landscape, diverse ecosystems, and a temperate climate. Poland is composed of Voivodeships of Poland, sixteen voivodeships and is the fifth most populous member state of the European Union (EU), with over 38 million people, and the List of European countries by area, fifth largest EU country by area, covering . The capital and List of cities and towns in Poland, largest city is Warsaw; other major cities include Kraków, Wrocław, Łódź, Poznań, and Gdańsk. Prehistory and protohistory of Poland, Prehistoric human activity on Polish soil dates to the Lower Paleolithic, with continuous settlement since the end of the Last Gla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
