 # Jeu

0-Knight's graph. Knight's tour graph is a knight's tour graph of an chessboard. For a knight's tour graph the total number of vertices is simply . For a and the total number of edges is A Hamiltonian path on the knight's tour graph is a knight's tour. Schwenk's theorem characterizes the sizes of chessboard for which a knight's tour exist. References See also 1-Tree (graph theory) The term "tree" was coined in 1857 by the British mathematician Arthur Cayley. A tree is an undirected simple graph G that satisfies any of the following equivalent conditions: If G has finitely many vertices, say n of them, then the above statements are also equivalent to any of the following conditions: G is connected and has n − 1 edges.G has no simple cycles and has n − 1 edges.

As elsewhere in graph theory, the order-zero graph (graph with no vertices) is generally excluded from consideration: while it is vacuously connected as a graph (any two vertices can be connected by a path), it is not 0-connected (or even (−1)-connected) in algebraic topology, unlike non-empty trees, and violates the "one more node than edges" relation. The term hedge sometimes refers to an ordered sequence of trees. In a rooted tree, the parent of a vertex is the vertex connected to it on the path to the root; every vertex except the root has a unique parent. An alternative proof uses Prüfer sequences. .) 2-Caterpillar tree. A caterpillar Caterpillars were first studied in a series of papers by Harary and Schwenk. The name was suggested by A. Hobbs. As Harary & Schwenk (1973) colorfully write, "A caterpillar is a tree which metamorphoses into a path when its cocoon of endpoints is removed. " Equivalent characterizations The following characterizations all describe the caterpillar trees: Generalizations A k-tree is a chordal graph with exactly n − k maximal cliques, each containing k + 1 vertices; in a k-tree that is not itself a (k + 1)-clique, each maximal clique either separates the graph into two or more components, or it contains a single leaf vertex, a vertex that belongs to only a single maximal clique.

Enumeration Caterpillars provide one of the rare graph enumeration problems for which a precise formula can be given: when n ≥ 3, the number of caterpillars with n unlabeled vertices is  For n = 1, 2, 3, ... the numbers of n-vertex caterpillars are Computational Complexity 3-List of graphs. For collected definitions of graph theory terms that do not refer to individual graph types, such as vertex and path, see Glossary of graph theory. For links to existing articles about particular kinds of graphs, see Category:Graphs. Gear A gear graph, denoted Gn is a graph obtained by inserting an extra vertex between each pair of adjacent vertices on the perimeter of a wheel graph Wn. Thus, Gn has 2n+1 vertices and 3n edges. Gear graphs are examples of squaregraphs, and play a key role in the forbidden graph characterization of squaregraphs. Gear graphs are also known as cogwheels and bipartite wheels.

Grid Helm A helm graph, denoted Hn is a graph obtained by attaching a single edge and node to each node of the outer circuit of a wheel graph Wn. Lobster Web The web graph W4,2 is a cube. A web graph has also been defined as a prism graph Yn+1, 3, with the edges of the outer cycle removed. See also Gallery of named graphs References 4-Wheel graph. Set-builder construction Given a vertex set of {1,2,3,…,v}, the edge set of the wheel graph can be represented in set-builder notation by {{1,2},{1,3},…,{1,v},{2,3},{3,4},…,{v-1,v},{v,2}}. Properties Wheel graphs are planar graphs, and as such have a unique planar embedding.

More specifically, every wheel graph is a Halin graph. There is always a Hamiltonian cycle in the wheel graph and there are cycles in Wn (sequence A002061 in OEIS). For odd values of n, Wn is a perfect graph with chromatic number 3: the vertices of the cycle can be given two colors, and the center vertex given a third color. The chromatic polynomial of the wheel graph Wn is : In matroid theory, two particularly important special classes of matroids are the wheel matroids and the whirl matroids, both derived from wheel graphs.

References 5-Hanan grid. Hanan grid generated for a 5-terminal case In geometry, the Hanan grid H(S) of a finite set S of points in the plane is obtained by constructing vertical and horizontal lines through each point in S. The main motivation for studying the Hanan grid stems from the fact that it is known to contain a rectilinear Steiner tree for S. It is named after M. Hanan, who was first to investigate the rectilinear Steiner minimum tree and introduced this graph. Jeux. Rook's graph. Definitions An n × m rook's graph represents the moves of a rook on an n × m chessboard. Its vertices may be given coordinates (x,y), where 1 ≤ x ≤ n and 1 ≤ y ≤ m. Two vertices (x1,y1) and (x2,y2) are adjacent if and only if either x1 = x2 or y1 = y2; that is, if they belong to the same rank or the same file of the chessboard.

For an n × m rook's graph the total number of vertices is simply nm. For an n × n rook's graph the total number of vertices is simply and the total number of edges is ; in this case the graph is also known as a two-dimensional Hamming graph or a Latin square graph. The rook's graph for an n × m chessboard may also be defined as the Cartesian product of two complete graphs Kn Km. Symmetry Rook's graphs are vertex-transitive and (n + m − 2)-regular; they are the only regular graphs formed from the moves of standard chess pieces in this way (Elkies).

Perfection The 3×3 rook's graph, colored with three colors and showing a clique of three vertices. Théorème flot-max/coupe-min. Un article de Wikipédia, l'encyclopédie libre. En optimisation, le théorème flot-max/coupe-min stipule qu'étant donné un graphe de flots, le flot maximum pouvant aller de la source au puits est égal à la capacité minimale devant être retirée du graphe afin d'empêcher qu'aucun flot ne puisse passer de la source au puits.

Ce théorème est un cas particulier du théorème de dualité en optimisation linéaire et généralise le théorème de Kőnig et le théorème de Hall (dans les graphes bipartis) et le théorème de Menger (dans les graphes quelconques). Définitions et notations[modifier | modifier le code] Soit un graphe orienté : On dénote par , la source, et le puits de .On associe à chaque arête de une capacité, qui représente le flot maximum pouvant passer par cette arête. Cette capacité est positive et on note , le vecteur dans contenant les valeurs de toutes les capacités.On associe aussi à chaque arête un flot représentant la quantité de flot passant par l'arête . On appelle coupe S-T de. X-Jeu de go.