Shannon Switching Game and Directed Variants

2015 ◽  
pp. 187-199
Author(s):  
A. P. Cláudio ◽  
S. Fonseca ◽  
L. Sequeira ◽  
I. P. Silva
Keyword(s):  
Switching Game

scholarly journals Maker–Breaker percolation games I: crossing grids

2020 ◽  
pp. 1-28
Author(s):  
A. Nicholas Day ◽  
Victor Falgas-Ravry
Keyword(s):  
Percolation Theory ◽  
Winning Strategy ◽  
List Type ◽  
Rectangular Grid ◽  
Grid Graph ◽  
Left Hand ◽  
Right Hand ◽  
Winning Strategies ◽  
Special Case ◽  
Switching Game

Abstract Motivated by problems in percolation theory, we study the following two-player positional game. Let Λm×n be a rectangular grid-graph with m vertices in each row and n vertices in each column. Two players, Maker and Breaker, play in alternating turns. On each of her turns, Maker claims p (as yet unclaimed) edges of the board Λm×n, while on each of his turns Breaker claims q (as yet unclaimed) edges of the board and destroys them. Maker wins the game if she manages to claim all the edges of a crossing path joining the left-hand side of the board to its right-hand side, otherwise Breaker wins. We call this game the (p, q)-crossing game on Λm×n. Given m, n ∈ ℕ, for which pairs (p, q) does Maker have a winning strategy for the (p, q)-crossing game on Λm×n? The (1, 1)-case corresponds exactly to the popular game of Bridg-it, which is well understood due to it being a special case of the older Shannon switching game. In this paper we study the general (p, q)-case. Our main result is to establish the following transition. If p ≥ 2q, then Maker wins the game on arbitrarily long versions of the narrowest board possible, that is, Maker has a winning strategy for the (2q, q)-crossing game on Λm×(q+1) for any m ∈ ℕ. If p ≤ 2q − 1, then for every width n of the board, Breaker has a winning strategy for the (p, q)-crossing game on Λm×n for all sufficiently large board-lengths m. Our winning strategies in both cases adapt more generally to other grids and crossing games. In addition we pose many new questions and problems.

scholarly journals Stochastic differential switching game in infinite horizon

2019 ◽  
Vol 474 (2) ◽  
pp. 793-813 ◽  
Author(s):  
Brahim El Asri ◽  
Sehail Mazid
Keyword(s):  
Infinite Horizon ◽  
Switching Game

scholarly journals The correct solution to Berlekamp's switching game

2004 ◽  
Vol 287 (1-3) ◽  
pp. 145-150 ◽  
Author(s):  
Jordan Carlson ◽  
Daniel Stolarski
Keyword(s):  
Correct Solution ◽  
Switching Game

A Solution of the Shannon Switching Game

1964 ◽  
Vol 12 (4) ◽  
pp. 687-725 ◽  
Author(s):  
Alfred Lehman
Keyword(s):  
Switching Game

scholarly journals The directed switching game on Lawrence oriented matroids

2009 ◽  
Vol 30 (8) ◽  
pp. 1833-1834 ◽  
Author(s):  
David Forge ◽  
Adrien Vieilleribière
Keyword(s):  
Oriented Matroids ◽  
Switching Game

Strategies for the Shannon Switching Game

1996 ◽  
Vol 103 (3) ◽  
pp. 250 ◽  
Author(s):  
Richard Mansfield
Keyword(s):  
Switching Game

scholarly journals Anti-Codes in Terms of Berlekamp's Switching Game

10.37236/17 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Projective Space ◽  
Linear Code ◽  
Coding Theory ◽  
Normal Forms ◽  
Light Pattern ◽  
Maximal Weight ◽  
Axis Parallel ◽  
Generalized Games ◽  
Switching Game ◽  
Zero Points

We view a linear code (subspace) $C\leq\mathbb{F}_{q}^n$ as a light pattern on the \(n\)-dimensional Berlekamp Board $\mathbb{F}_{q}^n$ with $q^n$ light bulbs. The lights corresponding to elements of $C$ are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. We show that the dual code $C^\perp$ contains a vector $v$ of full weight, i.e. $v_1,v_2,\dots,v_n\neq0$, if and only if the light pattern $C$ cannot be switched off. Generalizations of this allow us to describe anti-codes with maximal weight $\delta$ in a similar way, or, alternatively, in terms of a switching game in projective space. We provide convenient bases and normal forms to the modules of all light patterns of the generalized games. All our proofs are purely combinatorial and simpler than the algebraic ones used for similar results about anti-codes in $\mathbb{Z}_k^n$.  Aside from coding theory, the game is also of interest in the study of nowhere-zero points of matrices and nowhere-zero flows and colorings of graphs.

scholarly journals Some New Characterizations of Graph Colorability and of Blocking Sets of Projective Spaces

10.37236/3767 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Prime Power ◽  
Projective Spaces ◽  
Vertex Coloring ◽  
General Level ◽  
Blocking Sets ◽  
Proper Vertex ◽  
Switching Game

Let $G=(V,E)$ be a graph and $q$ be an odd prime power. We prove that $G$ possess a proper vertex coloring with $q$ colors if and only if there exists an odd vertex labeling $x\in F_q^V$ of $G$. Here, $x$ is called odd if there is an odd number of partitions $\pi=\{V_1,V_2,\dotsc,V_t\}$ of $V$ whose blocks $V_i$ are \(G\)-bipartite and \(x\)-balanced, i.e., such that $G|_{V_i}$ is connected and bipartite, and $\sum_{v\in V_i}x_v=0$. Other new characterizations concern edge colorability of graphs and, on a more general level, blocking sets of projective spaces. Some of these characterizations are formulated in terms of a new switching game.

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