Abstract
Shannon’s switching game is a combinatorial game invented by C. Shannon circa 1955 as a simple model for breakdown repair of the connectivity of a network. The game was completely solved by A. Lehman, shortly after, in what is considered the first application of matroid theory. In the middle 1980s Y. O. Hamidoune and M. Las Vergnas introduced and solved directed versions of the game for graphs considering their generalization to oriented matroids. We do a brief review of the main results and conjectures of the directed case.
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References
Berge, C.: Graphs. North-Holland Mathematical Library, vol. 6. North-Holland, Amsterdam (1989)
Berlekamp, E.R., Conway, J.H., Guy, R.K.: Winning Ways for Your Mathematical Plays, Volume II, 3rd ed. Academic Press, London (1985)
Björner, A., Las Vergnas, M., Sturmfels, B., White, N., Ziegler, G.: Oriented Matroids. Encyclopedia of Mathematics and Its Applications, vol. 46, 2nd edn. Cambridge University Press, Cambridge (1999)
Bland, R., Las Vergnas, M.: Orientability of matroids. J. Combin. Theory, Ser. B 24, 94–123 (1978)
Brualdi, R.: Introductory Combinatorics. North-Holland, Amsterdam (1977)
Bruno, J., Weinberg, L.: A constructive graph theoretic solution of the Shannon switching game. IEEE Trans. Circuit Theory CT-17(1), 74–81 (1970)
Bruno, J., Weinberg, L.: The principal minors of a matroid. Linear Algebra Appl. 4, 17–54 (1971)
Chatelain, V., Ramirez Alfonsin, J.L.: The switching game on unions of oriented matroids. Eur. J. Combin. 33(2), 215–219 (2012)
Cláudio, A.P., Fonseca, S., Sequeira, L., Silva, I.P.: Implementations of TREE and ARBORESCENCE: http://shlvgraphgame.fc.ul.pt/ (2014)
Forge, D., Vieilleribière, A.: The directed switching game on Lawrence Oriented matroids. Eur. J. Combin. 30(8), 1833–1834 (2009)
Gross, J.L., Yellen, J., Zhang, P. (eds.): Handbook of Graph Theory, 2nd ed, CRC Press 2004.
Hamidoune, Y.O., Las Vergnas, M.: Directed switching games on graphs and matroids. J. Combin. Theory, Ser. B 40, 237–269 (1986)
Hamidoune, Y.O., Las Vergnas, M.: Directed switching games II. Discret. Math. 165/166, 397–402 (1997)
Kishi, G., Kajitani, Y.: On maximally distant trees. In: Proceedings of 5th Allerton Conference on Circuits and Systems Theory, pp. 635–643 (1967)
Lehman, A.: A Solution to the Shannon switching game. J. Soc. Ind. Appl. Math. 12, 687–725 (1964)
Novak, L., Gibbons, A.: Hybrid Graph Theory and Network Analysis. Cambridge Tracts in Theoretical Computer Science, vol. 49. Cambridge University Press, Cambridge (1999)
Oxley, J.: Matroid Theory, 2nd ed. Oxford University Press, New York (2011)
Welsh, D.: Matroid Theory. Academic Press, London (1976)
Acknowledgements
This work was partially supported by Fundação para a Ciência e a Tecnologia, PEst-OE/MAT/UI0209/2013.
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Dedicated to the memory of Yahya Ould Hamidoune (1947–2011) and Michel las Vergnas (1941–2013)
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Cláudio, A.P., Fonseca, S., Sequeira, L., Silva, I.P. (2015). Shannon Switching Game and Directed Variants. In: Bourguignon, JP., Jeltsch, R., Pinto, A., Viana, M. (eds) Dynamics, Games and Science. CIM Series in Mathematical Sciences, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-16118-1_10
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