Fixed points in conjunctive networks and maximal independent sets in graph contractions

Given a graph G, viewed as a loop-less symmetric digraph, we study the maximum number of fixed points in a conjunctive boolean network with G as interaction graph. We prove that if G has no induced C4, then this quantity equals both the number of maximal independent sets in G and the maximum number of maximal independent sets among all the graphs obtained from G by contracting some edges. We also prove that, in the general case, it is coNP-hard to decide if one of these equalities holds, even if G has a unique induced C4.