3,659 research outputs found

    Characterising and recognising game-perfect graphs

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    Consider a vertex colouring game played on a simple graph with kk permissible colours. Two players, a maker and a breaker, take turns to colour an uncoloured vertex such that adjacent vertices receive different colours. The game ends once the graph is fully coloured, in which case the maker wins, or the graph can no longer be fully coloured, in which case the breaker wins. In the game gBg_B, the breaker makes the first move. Our main focus is on the class of gBg_B-perfect graphs: graphs such that for every induced subgraph HH, the game gBg_B played on HH admits a winning strategy for the maker with only Ο‰(H)\omega(H) colours, where Ο‰(H)\omega(H) denotes the clique number of HH. Complementing analogous results for other variations of the game, we characterise gBg_B-perfect graphs in two ways, by forbidden induced subgraphs and by explicit structural descriptions. We also present a clique module decomposition, which may be of independent interest, that allows us to efficiently recognise gBg_B-perfect graphs.Comment: 39 pages, 8 figures. An extended abstract was accepted at the International Colloquium on Graph Theory (ICGT) 201

    Positional games on random graphs

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    We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability pFp_{F} for the existence of Maker's strategy to claim a member of FF in the unbiased game played on the edges of random graph G(n,p)G(n,p), for various target families FF of winning sets. More generally, for each probability above this threshold we study the smallest bias bb such that Maker wins the (1β€…b)(1\:b) biased game. We investigate these functions for a number of basic games, like the connectivity game, the perfect matching game, the clique game and the Hamiltonian cycle game

    Guessing Numbers of Odd Cycles

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    For a given number of colours, ss, the guessing number of a graph is the base ss logarithm of the size of the largest family of colourings of the vertex set of the graph such that the colour of each vertex can be determined from the colours of the vertices in its neighbourhood. An upper bound for the guessing number of the nn-vertex cycle graph CnC_n is n/2n/2. It is known that the guessing number equals n/2n/2 whenever nn is even or ss is a perfect square \cite{Christofides2011guessing}. We show that, for any given integer sβ‰₯2s\geq 2, if aa is the largest factor of ss less than or equal to s\sqrt{s}, for sufficiently large odd nn, the guessing number of CnC_n with ss colours is (nβˆ’1)/2+log⁑s(a)(n-1)/2 + \log_s(a). This answers a question posed by Christofides and Markstr\"{o}m in 2011 \cite{Christofides2011guessing}. We also present an explicit protocol which achieves this bound for every nn. Linking this to index coding with side information, we deduce that the information defect of CnC_n with ss colours is (n+1)/2βˆ’log⁑s(a)(n+1)/2 - \log_s(a) for sufficiently large odd nn. Our results are a generalisation of the s=2s=2 case which was proven in \cite{bar2011index}.Comment: 16 page

    The Computational Complexity of the Game of Set and its Theoretical Applications

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    The game of SET is a popular card game in which the objective is to form Sets using cards from a special deck. In this paper we study single- and multi-round variations of this game from the computational complexity point of view and establish interesting connections with other classical computational problems. Specifically, we first show that a natural generalization of the problem of finding a single Set, parameterized by the size of the sought Set is W-hard; our reduction applies also to a natural parameterization of Perfect Multi-Dimensional Matching, a result which may be of independent interest. Second, we observe that a version of the game where one seeks to find the largest possible number of disjoint Sets from a given set of cards is a special case of 3-Set Packing; we establish that this restriction remains NP-complete. Similarly, the version where one seeks to find the smallest number of disjoint Sets that overlap all possible Sets is shown to be NP-complete, through a close connection to the Independent Edge Dominating Set problem. Finally, we study a 2-player version of the game, for which we show a close connection to Arc Kayles, as well as fixed-parameter tractability when parameterized by the number of rounds played

    Shapley Meets Shapley

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    This paper concerns the analysis of the Shapley value in matching games. Matching games constitute a fundamental class of cooperative games which help understand and model auctions and assignments. In a matching game, the value of a coalition of vertices is the weight of the maximum size matching in the subgraph induced by the coalition. The Shapley value is one of the most important solution concepts in cooperative game theory. After establishing some general insights, we show that the Shapley value of matching games can be computed in polynomial time for some special cases: graphs with maximum degree two, and graphs that have a small modular decomposition into cliques or cocliques (complete k-partite graphs are a notable special case of this). The latter result extends to various other well-known classes of graph-based cooperative games. We continue by showing that computing the Shapley value of unweighted matching games is #P-complete in general. Finally, a fully polynomial-time randomized approximation scheme (FPRAS) is presented. This FPRAS can be considered the best positive result conceivable, in view of the #P-completeness result.Comment: 17 page

    Positional Games

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    Positional games are a branch of combinatorics, researching a variety of two-player games, ranging from popular recreational games such as Tic-Tac-Toe and Hex, to purely abstract games played on graphs and hypergraphs. It is closely connected to many other combinatorial disciplines such as Ramsey theory, extremal graph and set theory, probabilistic combinatorics, and to computer science. We survey the basic notions of the field, its approaches and tools, as well as numerous recent advances, standing open problems and promising research directions.Comment: Submitted to Proceedings of the ICM 201
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