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

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In this article we study the minimum number κ 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 NECC graph built from the BAN and the update schedule. We show the relation between κ and the chromatic number of the NECC graph. Thanks to this NECC graph, we bound κ 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 NECC graph is always less than or equal to n/2 and, for the subclass of bijective BANs, κ is always less than or equal to n/2+1.