Fixed points and maximal independent sets in AND-OR networks

We study the maximum number of 2xed points of boolean networks with local update function AND–OR. We prove that this number for networks with connected digraph is 2(n−1)=2 for n odd and 2(n−2)=2 + 1 for n even if the digraph has not loops; and 2n−1 + 1 otherwise, where n is the number of nodes of the digraph. We also exhibit some networks reaching these bounds. To obtain these results we construct a bijection between the maximal independent sets of the digraph and the 2xed points of the network belonging to a particular family of AND–OR networks. ? 2003 Elsevier B.V. All rights reserved.