Irreversible Dynamos in Tori

On time versus size for monotone dynamic monopolies in regular topologies

We consider a well known distributed coloring game played on a simple connected graph: initially, each vertex is colored black or white; at each round, each vertex simultaneously recolors itself by the color of the simple (strong) majority of its neighbours. A set of vertices is said to be a dynamo, if starting the game with only the vertices of colored black, the computation eventually reaches...

Irreversible Dynamos in Butterflies

We study the propagation of information in a network in the presence of permanent faults, to detect the patterns of initial faults which may lead the entire system to fail. Such patterns, called dynamos, have been already studied in diierent contexts and topologies, and under diierent laws of fault propagation. In our model each node assumes a new state according to the majority of the states o...

Optimal Irreversible Dynamos in Chordal Rings

Bounding the Number of Tolerable Faults in Majority-Based Systems

Consider the following coloring process in a simple directed graph G(V,E) with positive indegrees. Initially, a set S of vertices are white. Thereafter, a black vertex is colored white whenever the majority of its in-neighbors are white. The coloring process ends when no additional vertices can be colored white. If all vertices end up white, we call S an irreversible dynamic monopoly (or dynamo...

Omega Polynomial in Polybenzene Multi Tori

The polybenzene units BTX 48, X=A (armchair) and X=Z (zig-zag) dimerize forming “eclipsed” isomers, the oligomers of which form structures of five-fold symmetry, called multi-tori. Multi-tori can be designed by appropriate map operations. The genus of multi-tori was calculated from the number of tetrapodal units they consist. A description, in terms of Omega polynomial, of the two linearly peri...