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Bishop's Graph
In mathematics, a bishop's graph is a graph that represents all legal moves of the chess piece the bishop on a chessboard. Each vertex represents a square on the chessboard and each edge represents a legal move of the bishop; that is, there is an edge between two vertices (squares) if they occupy a common diagonal. When the chessboard has dimensions m\times n, then the induced graph is called the m\times n bishop's graph. Properties The fact that the chessboard has squares of two colors, say red and black, such that squares that are horizontally or vertically adjacent have opposite colors, implies that the bishop's graph has two connected components, whose vertex sets are the red and the black squares, respectively. The reason is that the bishop's diagonal moves do not allow it to change colors, but by one or more moves a bishop can get from any square to any other of the same color. The two components are isomorphic if the board has a side of even length, but not if both side ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a Set (mathematics), set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called ''Vertex (graph theory), 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. Graphs are one of the objects of study in discrete mathematics. 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'' m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chess Piece
A chess piece, or chessman, is a game piece that is placed on a chessboard to play the game of chess. It can be either White and Black in chess, white or black, and it can be one of six types: King (chess), king, Queen (chess), queen, Rook (chess), rook, Bishop (chess), bishop, Knight (chess), knight, or Pawn (chess), pawn. Chess sets generally come with sixteen pieces of each color. Additional pieces, usually an extra queen per color, may be provided for use in Promotion (chess), promotion. Number of pieces Each player begins with sixteen pieces (but see the #Usage of the term piece, subsection below for other usage of the term ''piece''). The pieces that belong to each player are distinguished by color: the lighter colored pieces are referred to as "white" and the player that owns them as "White", whereas the darker colored pieces are referred to as "black" and the player that owns them as "Black". In a standard game, each of the two players begins with the following sixteen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bishop (chess)
The bishop (♗, ♝) is a piece in the game of chess. It moves and captures along without jumping over intervening pieces. Each player begins the game with two bishops. One starts between the and the king, the other between the and the queen. The starting squares are c1 and f1 for White's bishops, and c8 and f8 for Black's bishops. Placement and movement The king's bishop is placed between the king and the king's knight, f1 for White and f8 for Black; the queen's bishop is placed between the queen and the queen's knight, c1 for White and c8 for Black. The bishop has no restrictions in distance for each move but is limited to diagonal movement. It cannot jump over other pieces. A bishop captures by occupying the square on which an enemy piece stands. As a consequence of its diagonal movement, each bishop always remains on one square color. Due to this, it is common to refer to a bishop as a light-squared or dark-squared bishop. Comparison – other pieces Versus rook A r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chessboard
A chessboard is a used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the board is oriented such that each player's near-right corner square is a light square. The columns of a chessboard are known as ', the rows are known as ', and the lines of adjoining same-coloured squares (each running from one edge of the board to an adjacent edge) are known as '. Each square of the board is named using algebraic, descriptive, or numeric chess notation; algebraic notation is the FIDE standard. In algebraic notation, using White's perspective, files are labeled ''a'' through ''h'' from left to right, and ranks are labeled ''1'' through ''8'' from bottom to top; each square is identified by the file and rank which it occupies. The a- through d-files comprise the , while the e- through h-files comprise the . History and evo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (graph Theory)
In discrete mathematics, and more specifically in graph theory, a vertex (plural vertices) or node is the fundamental unit of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges (unordered pairs of vertices), while a directed graph consists of a set of vertices and a set of arcs (ordered pairs of vertices). In a diagram of a graph, a vertex is usually represented by a circle with a label, and an edge is represented by a line or arrow extending from one vertex to another. From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects. The two vertices forming an edge are said to be the endpoints of this edge, and the edge is said to be incident to the vertices. A vertex ''w'' is said to be ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge (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 K L M N O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rook's Graph
In graph theory, a rook's graph is a graph that represents all legal moves of the rook chess piece on a chessboard. Each vertex of a rook's graph represents a square on a chessboard, and each edge connects two squares on the same row (rank) or on the same column (file) as each other, the squares that a rook can move between. These graphs can be constructed for chessboards of any rectangular shape, and can be defined mathematically as the Cartesian product of two complete graphs, as the two-dimensional Hamming graphs, or as the line graphs of complete bipartite graphs. Rook's graphs are highly symmetric, having symmetries taking every vertex to every other vertex. In rook's graphs defined from square chessboards, more strongly, every two edges are symmetric, and every pair of vertices is symmetric to every other pair at the same distance (they are distance-transitive). For chessboards with relatively prime dimensions, they are circulant graphs. With one exception, they can be dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rook (chess)
The rook (; ♖, ♜) is a piece in the game of chess. It may move any number of squares horizontally or vertically without jumping, and it may an enemy piece on its path; additionally, it may participate in castling. Each player starts the game with two rooks, one in each corner on their own side of the board. Formerly, the rook (from Persian رخ ''rokh''/''rukh'', meaning "chariot") was alternatively called the tower, marquess, rector, and comes (count or earl). The term "castle" is considered to be informal, incorrect, or old-fashioned. Placement and movement The white rooks start on squares a1 and h1, while the black rooks start on a8 and h8. The rook moves horizontally or vertically, through any number of unoccupied squares (see diagram). The rook cannot jump over pieces. The rook may capture an enemy piece by moving to the square on which the enemy piece stands, removing it from play. The rook also participates with the king in a special move called castling, wherein i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dominating Set
In graph theory, a dominating set for a graph is a subset of its vertices, such that any vertex of is either in , or has a neighbor in . The domination number is the number of vertices in a smallest dominating set for . The dominating set problem concerns testing whether for a given graph and input ; it is a classical NP-complete decision problem in computational complexity theory. Therefore it is believed that there may be no efficient algorithm that can compute for all graphs . However, there are efficient approximation algorithms, as well as efficient exact algorithms for certain graph classes. Dominating sets are of practical interest in several areas. In wireless networking, dominating sets are used to find efficient routes within ad-hoc mobile networks. They have also been used in document summarization, and in designing secure systems for electrical grids. Formal definition Given an undirected graph , a subset of vertices D\subseteq V is called a dominating se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 connected by '' edges'' (also called ''links'' or ''lines''). A distinction is made between undirected graphs, where edges link two vertices symmetrically, and directed graphs, where edges link two vertices asymmetrically. Graphs are one of the principal objects of study in discrete mathematics. Definitions Definitions in graph theory vary. The following are some of the more basic ways of defining graphs and related mathematical structures. Graph In one restricted but very common sense of the term, a graph is an ordered pair G=(V,E) comprising: * V, a set of vertices (also called nodes or points); * E \subseteq \, a set of edges (also called links or lines), which are unordered pairs of vertices (that is, an edge is associated with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Chess Problems
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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