On the cost of simulating a parallel Boolean automata network with a block-sequential one

arXiv:1702.03101International audienceIn this article we study the minimum number $\kappa$ of additional automata that a Boolean automata network (BAN) associated with a given block-sequential update schedule needs in order to simulate a given BAN with a parallel update schedule. We introduce a graph that we call $\GNECC$ graph built from the BAN and the update schedule. We show the relation between $\kappa$ and the chromatic number of the $\GNECC$ graph. Thanks to this $\GNECC$ graph, we bound $\kappa$ in the worst case between $n/2$ and $2n/3+2$ ($n$ being the size of the BAN simulated) and we conjecture that this number equals $n/2$. We support this conjecture with two results: the clique number of a $\GNECC$ graph is always less than or equal to $n/2$ and, for the subclass of bijective BANs, $\kappa$ is always less than or equal to $n/2+1$