Pursuit–evasion

	Pursuit–evasion Pursuit–evasion (variants of which are referred to as cops and robbers and graph searching) is a family of problems in mathematics and computer science in which one group attempts to track down members of another group in an environment. Early work on problems of this type modeled the environment geometrically. In 1976, Torrence Parsons introduced a formulation whereby movement is constrained by a graph. The geometric formulation is sometimes called continuous pursuit–evasion, and the graph formulation discrete pursuit–evasion (also called graph searching). Current research is typically limited to one of these two formulations. Discrete formulation In the discrete formulation of the pursuit–evasion problem, the environment is modeled as a graph. Problem definition There are innumerable possible variants of pursuit–evasion, though they tend to share many elements. A typical, basic example is as follows (cops and robber games): Pursuers and evaders occupy nodes of a gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Angel Problem The angel problem is a question in combinatorial game theory proposed by John Horton Conway. The game is commonly referred to as the angels and devils game.John H. Conway, The angel problem', in: Richard Nowakowski (editor) ''Games of No Chance'', volume 29 of MSRI Publications, pages 3–12, 1996. The game is played by two players called the angel and the devil. It is played on an infinite chessboard (or equivalently the points of a 2D lattice). The angel has a power ''k'' (a natural number 1 or higher), specified before the game starts. The board starts empty with the angel in one square. On each turn, the angel jumps to a different empty square which could be reached by at most ''k'' moves of a chess king, i.e. the distance from the starting square is at most ''k'' in the infinity norm. The devil, on its turn, may add a block on any single square not containing the angel. The angel may leap over blocked squares, but cannot land on them. The devil wins if the angel is u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Pursuit Curve In geometry, a curve of pursuit is a curve constructed by analogy to having a point (geometry), point or points representing pursuers and pursuees; the curve of pursuit is the curve traced by the pursuers. Definition With the paths of the pursuer and pursuee Parametrization (geometry), parameterized in time, the pursuee is always on the pursuer's tangent. That is, given , the pursuer (follower), and , the pursued (leader), for every with there is an such that :L(t) = F(t) + xF'\!(t). History The pursuit curve was first studied by Pierre Bouguer in 1732. In an article on navigation, Bouguer defined a curve of pursuit to explore the way in which one ship might maneuver while pursuing another. Leonardo da Vinci has occasionally been credited with first exploring curves of pursuit. However Paul J. Nahin, having traced such accounts as far back as the late 19th century, indicates that these anecdotes are unfounded. Single pursuer The path followed by a single pursuer, followi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Haven (graph Theory) In graph theory, a haven is a certain type of function on sets of vertices in an undirected graph. If a haven exists, it can be used by an evader to win a pursuit–evasion game on the graph, by consulting the function at each step of the game to determine a safe set of vertices to move into. Havens were first introduced by as a tool for characterizing the treewidth of graphs. Their other applications include proving the existence of small separators on minor-closed families of graphs, and characterizing the ends and clique minors of infinite graphs... Definition If is an undirected graph, and is a set of vertices, then an -flap is a nonempty connected component of the subgraph of formed by deleting . A haven of order in is a function that assigns an -flap to every set of fewer than vertices. This function must also satisfy additional constraints which are given differently by different authors. The number is called the ''order'' of the haven.. In the original ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Cop-win Graph In graph theory, a cop-win graph is an undirected graph on which the pursuer (cop) can always win a pursuit–evasion game against a robber, with the players taking alternating turns in which they can choose to move along an edge of a graph or stay put, until the cop lands on the robber's vertex. Finite cop-win graphs are also called dismantlable graphs or constructible graphs, because they can be dismantled by repeatedly removing a dominated vertex (one whose Neighbourhood (graph theory), closed neighborhood is a subset of another vertex's neighborhood) or constructed by repeatedly adding such a vertex. The cop-win graphs can be recognized in polynomial time by a greedy algorithm that constructs a dismantling order. They include the chordal graphs, and the graphs that contain a universal vertex. Definitions Pursuit–evasion Cop-win graphs can be defined by a pursuit–evasion game in which two players, a cop and a robber, are positioned at different initial vertices of a given u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]