The number of fixed points of the majority rule
Discrete Mathematics
On a paper of Agur, Fraenkel and Klein
Discrete Mathematics
The r-majority vote action on 0-1 sequences
Discrete Mathematics
Parametrization for stationary patterns of the r-majority operators on 0-1 sequences
Discrete Mathematics
Distributed loop computer networks: a survey
Journal of Parallel and Distributed Computing
Testing and reconfiguration of VLSI linear arrays
Theoretical Computer Science
Optimal irreversible dynamos in chordal rings
Discrete Applied Mathematics - special issue on the 25th international workshop on graph theoretic concepts in computer science (WG'99)
On the Complexity of Testing for Catastrophic Faults
ISAAC '95 Proceedings of the 6th International Symposium on Algorithms and Computation
Euro-Par '98 Proceedings of the 4th International Euro-Par Conference on Parallel Processing
On time versus size for monotone dynamic monopolies in regular topologies
Journal of Discrete Algorithms
Global defensive alliances in star graphs
Discrete Applied Mathematics
Dynamic monopolies with randomized starting configuration
Theoretical Computer Science
Dynamic monopolies and feedback vertex sets in hexagonal grids
Computers & Mathematics with Applications
Global defensive alliances of trees and Cartesian product of paths and cycles
Discrete Applied Mathematics
Fault recovery in wireless networks: the geometric recolouring approach
SEA'10 Proceedings of the 9th international conference on Experimental Algorithms
Bounding the number of tolerable faults in majority-based systems
CIAC'10 Proceedings of the 7th international conference on Algorithms and Complexity
Stable sets of threshold-based cascades on the erdős-rényi random graphs
IWOCA'11 Proceedings of the 22nd international conference on Combinatorial Algorithms
Discrete Applied Mathematics
On the non-progressive spread of influence through social networks
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Triggering cascades on undirected connected graphs
Information Processing Letters
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
Bounding the sizes of dynamic monopolies and convergent sets for threshold-based cascades
Theoretical Computer Science
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Let G be a simple connected graph where every node is colored either black or white. Consider now the following repetitive process on G: each node recolors itself, at each local time step, with the color held by the majority of its neighbors. Depending on the initial assignment of colors to the nodes and on the definition of majority, different dynamics can occur. We are interested in dynamos; i.e., initial assignments of colors which lead the system to a monochromatic configuration in a finite number of steps. In the context of distributed computing and communication networks, this repetitive process is particularly important in that it describes the impact that a set of initial faults can have in majority-based systems (where black nodes correspond to faulty elements and white to non-faulty ones). In this paper, we study two particular forms of dynamos (irreversible and monotone) in tori, focusing on the minimum number of initial black elements needed to reach the fixed point. We derive lower and upper bounds on the size of dynamos for three types of tori, under different assumptions on the majority rule (simple and strong). These bounds are tight within an additive constant. The upper bounds are constructive: for each topology and each majority rule, we exhibit a dynamo of the claimed size.