# The firing squad synchronization problem on Cayley graphs

## Abstract

The Firing Squad Synchronization Problem (FSSP for short) has been intensively studied in the one-dimensional space. The problem consists in the synchronization of a segment of automata. We generalize this problem on Cayley graphs. We give minimal time solutions for (*a*) synchronizing all cells in all minimal paths between any pair of cells of a Cayley graph; (*b*) synchronizing all cells in all minimal paths starting at a given cell *G* (the “general”) and leading to all cells at a given distance from *G* in a Cayley graph. In solutions for (b), in some cases, all cells of a ball in a Cayley graph will be synchronized, in other cases this is not possible because of the existence of “culs-de-sac”.

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