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Grid Graph
In graph theory, a lattice graph, mesh graph, or grid graph is a Graph (discrete mathematics), graph whose graph drawing, drawing, Embedding, embedded in some Euclidean space , forms a regular tiling. This implies that the group (mathematics), group of Bijection, bijective transformations that send the graph to itself is a lattice (group), lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid". The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of graphs, Cartesian product of a number of complete graphs. Square grid graph A comm ...
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Bipartite Graph
In the mathematics, mathematical field of graph theory, a bipartite graph (or bigraph) is a Graph (discrete mathematics), graph whose vertex (graph theory), vertices can be divided into two disjoint sets, disjoint and Independent set (graph theory), independent sets U and V, that is, every edge (graph theory), edge connects a Vertex (graph theory), vertex in U to one in V. Vertex sets U and V are usually called the ''parts'' of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycle (graph theory), cycles. The two sets U and V may be thought of as a graph coloring, coloring of the graph with two colors: if one colors all nodes in U blue, and all nodes in V red, each edge has endpoints of differing colors, as is required in the graph coloring problem.. In contrast, such a coloring is impossible in the case of a non-bipartite graph, such as a Gallery of named graphs, triangle: after one node is colored blue and another red, the third vertex ...
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Lattice Path
In combinatorics, a lattice path in the -dimensional integer lattice of length with steps in the Set (mathematics), set , is a sequence of Vector (mathematics and physics), vectors such that each consecutive difference v_i - v_ lies in . A lattice path may lie in any Lattice (group), lattice in , but the integer lattice is most commonly used. An example of a lattice path in of length 5 with steps in S = \lbrace (2,0), (1,1), (0,-1) \rbrace is L = \lbrace (-1,-2), (0,-1), (2,-1), (2,-2), (2,-3), (4,-3) \rbrace . North-East lattice paths A North-East (NE) lattice path is a lattice path in \mathbb^2 with steps in S = \lbrace (0,1), (1,0) \rbrace . The (0,1) steps are called North steps and denoted by N s; the (1,0) steps are called East steps and denoted by E s. NE lattice paths most commonly begin at the origin. This convention allows encoding all the information about a NE lattice path L in a single permutation pattern, permutation word. The length of the wor ...
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Wazir (chess)
The wazir or vazir is a fairy chess piece that may move a single square vertically or horizontally. In notation, it is given the symbol ''W''. In this article, the wazir is represented by an inverted rook. Name etymology The name wazīr (vazir) (Arabic/Persian: وزير from Middle Persian vichir) means "minister" in several West and South Asian languages and is found in English as vizier. Wazīr (Vazir) is also the name of the queen in Arabic, Persian, and Hindi. History and nomenclature The wazir is a very old piece, appearing in some very early chess variants, such as Tamerlane chess. The wazir also appears in some historical large shogi variants, such as in dai shogi under the name ''angry boar'' (嗔猪 ''shinchō''). The general in xiangqi moves like a wazir but may not leave its palace or end its turn in check (chess), check. Value The wazir by itself is not much more powerful than a pawn (chess), pawn, but as an additional power to other pieces, it is worth about half a ...
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Fairy Chess Piece
A fairy chess piece, variant chess piece, unorthodox chess piece, or heterodox chess piece is a chess piece not used in conventional chess but incorporated into certain chess variants and some unorthodox chess problems, known as fairy chess. Compared to conventional pieces, fairy pieces vary mostly in Rules of chess#Movement, the way they move, but they may also follow special rules for capturing, promotions, etc. Because of the distributed and uncoordinated nature of unorthodox chess development, the same piece can have different names, and different pieces can have the same name in various contexts. Most are symbolised as inverted or rotated icons of the standard pieces in diagrams, and the meanings of these "wildcards" must be defined in each context separately. Pieces invented for use in chess variants rather than problems sometimes instead have special icons designed for them, but with some exceptions (the princess (chess), princess, empress (chess), empress, and occasionally a ...
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Chessboard
A chessboard is a game board 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 constitute the , and the e- through h-files constitute the ; the ...
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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 or handicap games. Number Each player begins with sixteen pieces (but see the #Definitions, 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 controls them as "White", whereas the darker colored pieces are referred to as "black" and the player that controls them as "Black". In a standard game, each of the two players begins with the following si ...
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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; it may participate in castling. Each player starts the game with two rooks, one in each corner on their side of the board. Formerly, the rook (from ) was alternatively called the ''tower'', ''marquess'', ''rector'', and ''comes'' (''count'' or ''earl''). The term "castle" is considered to be informal or old-fashioned. Placement and movement The white rooks start on the 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. 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 it is transferred to the square crossed by the king after th ...
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Rook's Graph
In graph theory, a rook's graph is an undirected 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 there is an edge between any two squares sharing a row (rank) or column (file), the squares that a rook can move between. These graphs can be constructed for chessboards of any rectangular shape. Although rook's graphs have only minor significance in chess lore, they are more important in the abstract mathematics of graphs through their alternative constructions: rook's graphs are the Cartesian product of two complete graphs, and are the line graphs of complete bipartite graphs. The square rook's graphs constitute the two-dimensional Hamming 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 ...
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Hanan Grid
In geometry, the Hanan grid of a finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. Th ... of points in the plane is obtained by constructing vertical and horizontal lines through each point in . The main motivation for studying the Hanan grid stems from the fact that it is known to contain a minimum length rectilinear Steiner tree for . It is named after Maurice Hanan, who was first to investigate the rectilinear Steiner minimum tree and introduced this graph.M. HananOn Steiner's problem with rectilinear distance, J. SIAM Appl. Math. 14 (1966), 255 - 265. References {{reflist Graph families Geometric graphs ...
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Bidimensionality
Bidimensionality theory characterizes a broad range of graph problems (bidimensional) that admit efficient approximate, fixed-parameter or kernel solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, bounded-genus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the graph minor theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. The theory was introduced in the work of Demaine, Fomin, Hajiaghayi, and Thilikos, for which the authors received the Nerode Prize in 2015. Definition A parameterized problem \Pi is a subset of \Gamma^\times \mathbb for some finite alphabet \Gamma. An instance of a parameterized problem consists of ''(x,k)'', where ''k'' is called the parameter. A parameterized problem \Pi is ''minor-bidimensional'' if # For any pair of graphs H,G, such that H is a minor of G and integer k, (G,k)\in \Pi yields that (H,k)\in ...
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Treewidth
In graph theory, the treewidth of an undirected graph is an integer number which specifies, informally, how far the graph is from being a tree. The smallest treewidth is 1; the graphs with treewidth 1 are exactly the trees and the forests A forest is an ecosystem characterized by a dense community of trees. Hundreds of definitions of forest are used throughout the world, incorporating factors such as tree density, tree height, land use, legal standing, and ecological functio .... An example of graphs with treewidth at most 2 are the series–parallel graphs. The maximal graphs with treewidth exactly are called '' -trees'', and the graphs with treewidth at most are called '' partial -trees''. Many other well-studied graph families also have bounded treewidth. Treewidth may be formally defined in several equivalent ways: in terms of the size of the largest vertex set in a tree decomposition of the graph, in terms of the size of the largest clique in a chordal completi ...
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