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Cop and Robber

Cop and Robber

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One way to prove this is to use subgraphs that are guardable by a single cop: the cop can move to track the robber in such a way that, if the robber ever moves into the subgraph, the cop can immediately capture the robber. However, the problems of obtaining a tight bound, and of proving or disproving Meyniel's conjecture, remain unsolved. a b c d e f g h i Nowakowski, Richard; Winkler, Peter (1983), "Vertex-to-vertex pursuit in a graph", Discrete Mathematics, 43 (2–3): 235–239, doi: 10.

Chepoi, Victor (1997), "Bridged graphs are cop-win graphs: an algorithmic proof", Journal of Combinatorial Theory, Series B, 69 (1): 97–100, doi: 10. It constructs and maintains the actual deficit set (neighbors of x that are not neighbors of y) only for pairs ( x, y) for which the deficit is small. September 2021), "Computability and the game of cops and robbers on graphs", Archive for Mathematical Logic, 61 (3–4): 373–397, doi: 10. These are graphs defined from the vertices of a polygon, with an edge whenever two vertices can be connected by a line segment that does not pass outside the polygon.

Gavenčiak, Tomáš (2010), "Cop-win graphs with maximum capture-time", Discrete Mathematics, 310 (10–11): 1557–1563, doi: 10. For other types of graphs, there may exist infinite cop-win graphs of that type even when there are no finite ones; for instance, this is true for the vertex-transitive graphs that are not complete graphs. The Moore bound in the degree diameter problem implies that at least one of these two kinds of guardable sets has size Ω ( log ⁡ n / log ⁡ log ⁡ n ) {\displaystyle \Omega (\log n/\log \log n)} .

Arboricity, h-index, and dynamic algorithms", Theoretical Computer Science, 426–427: 75–90, arXiv: 1005. Create blocks from an arbitrary partition of the vertices, and find the numbers representing the neighbors of each vertex in each block. A suitable distance away from the 'Cops' area, mark out a home base for the 'Robbers', where players will start out. Two types of subgraph that are guardable are the closed neighborhood of a single vertex, and a shortest path between any two vertices. A similar game with larger numbers of cops can be used to define the cop number of a graph, the smallest number of cops needed to win the game.Then, while staying in pairs whose first component is the same as the robber, the cop can play to win in the second of the two factors. However, there exist infinite chordal graphs, and even infinite chordal graphs of diameter two, that are not cop-win. If there is only one cop, the robber can move to a position two steps away from the cop, and always maintain the same distance after each move of the robber.

On bridged graphs and cop-win graphs", Journal of Combinatorial Theory, Series B, 44 (1): 22–28, doi: 10.The cop can win in a strong product of two cop-win graphs by, first, playing to win in one of these two factor graphs, reaching a pair whose first component is the same as the robber.

In the case of infinite graphs, it is possible to construct computable countably infinite graphs, on which an omniscient robber could always evade any cop, but for which no algorithm can follow this strategy.Your friends, classmates, colleagues or anyone else around the world, if you want, add him/her to your friends list. For, in a graph with no dominated vertices, if the robber has not already lost, then there is a safe move to a position not adjacent to the cop, and the robber can continue the game indefinitely by playing one of these safe moves at each turn. On the first turn of the game, the player controlling the cops places each cop on a vertex of the graph (allowing more than one cop to be placed on the same vertex). This is not true for all cop-win graphs; for instance, the five-vertex wheel graph is cop-win but contains an isometric 4-cycle, which is not cop-win, so this wheel graph is not hereditarily cop-win.

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