381 research outputs found

    An Algorithmic Analysis of the Honey-Bee Game

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    The Honey-Bee game is a two-player board game that is played on a connected hexagonal colored grid or (in a generalized setting) on a connected graph with colored nodes. In a single move, a player calls a color and thereby conquers all the nodes of that color that are adjacent to his own current territory. Both players want to conquer the majority of the nodes. We show that winning the game is PSPACE-hard in general, NP-hard on series-parallel graphs, but easy on outerplanar graphs. In the solitaire version, the goal of the single player is to conquer the entire graph with the minimum number of moves. The solitaire version is NP-hard on trees and split graphs, but can be solved in polynomial time on co-comparability graphs.Comment: 20 pages, 9 figure

    Hardness of Games and Graph Sampling

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    The work presented in this document is divided into two parts. The �rst part presents the hardness of games and the second part presents Graph sampling. Non-deterministic constraint logic[1] is used to prove the hardness of games. The games which are considered in this work is Reversi (2 player bounded game), Peg Solitaire (single player bounded game), Badland (single player bounded game). It also contains a theoretical study of peg solitaire on special graph classes. Reversi is proved to be PSPACE-Complete using Bounded 2CL, Peg Solitaire is proved to be NP-Complete using Bounded NCL. Badland is proved to be NP-Complete by a reduction from 3-SAT. The objective of study of peg solitaire of special graph classes is to �nd the maximum number of marbles we can remove from a fully �lled board, if the player is given the privilege to remove a marble from any cell initially, then following the rules after the initial move. The second part of the work is dedicated to graph sampling. Given a graph G, we try to sample a represen- tative subgraph Gs which is similar to the original graph G. The properties that are being studied are Degree Distribution, Clustering Coefficient, Average Shortest Path Length, Largest Connected Component Size. To measure the similarity between the original graph and sample we use the metrics Kolmogorov - Smirnov test and Kullback - Leibler divergence test. Tightly Induced Edge Sampling performs well on general graphs but it's performance decreases when the graph is a tree. Overall TIBFS and KARGER produces a sample which closely matches the distribution of original graphs.

    Endgame problems of Sim-like graph Ramsey avoidance games are PSPACE-complete

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    AbstractIn Sim, two players compete on a complete graph of six vertices (K6). The players alternate in coloring one as yet uncolored edge using their color. The player who first completes a monochromatic triangle (K3) loses. Replacing K6 and K3 by arbitrary graphs generalizes Sim to graph Ramsey avoidance games. Given an endgame position in these games, the problem of deciding whether the player who moves next has a winning strategy is shown to be PSPACE-complete. It can be reduced from the problem of whether the first player has a winning strategy in the game Gpos(POSCNF) (Schaefer, J. Comput. System Sci. 16 (2) (1978) 185–225). The following game variants are also shown to have PSPACE-complete endgame problems: (1) completing a monochromatic subgraph isomorphic to A is forbidden and the player who is first unable to move loses, (2) both players are allowed to color one or more edges in each move, (3) more than two players take part in the game, and (4) each player has to avoid a separate graph. In all results, the graphs to be avoided can be restricted to the bowtie graph (⋈, i.e., two triangles with one common vertex)

    A Graph Theoretical Approach to the Dollar Game Problem

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    In this thesis we consider a problem in Graph Theory known as the Dollar Game. The Dollar game was first introduced by Matthew Baker of the Georgia Institute of Technology in 2010. It is a game of solitaire, played on a graph, and is a variation of chip firing, or sand-piling games. Baker approached the problem within the context of Algebraic Geometry. It is the goal of this paper to provide an overview of the necessary graph theory to understand the problem presented in this game, as well as background on chip firing games, their history and evolution. Finally we will present a variety of results about the Dollar Game from a graph theoretical standpoint

    Extremal Results for Peg Solitaire on Graphs

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    In a 2011 paper by Beeler and Hoilman, the game of peg solitaire is generalized to arbitrary boards. These boards are treated as graphs in the combinatorial sense. An open problem from that paper is to determine the minimum number of edges necessary for a graph with a fixed number of vertices to be solvable. This thesis provides new bounds on this number. It also provides necessary and sufficient conditions for two families of graphs to be solvable, along with criticality results, and the maximum number of pegs that can be left in each of the two graph families
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