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

TL;DR

This paper explores the relationship between fixed points in conjunctive networks and maximal independent sets in graphs, establishing exact counts under certain conditions and proving computational hardness in the general case.

Contribution

It characterizes the maximum number of fixed points in conjunctive networks in terms of maximal independent sets and proves coNP-hardness of related decision problems.

Findings

If G has no induced C4, fixed points equal maximal independent sets

Maximum fixed points correspond to maximal independent sets in G

Deciding these equalities is coNP-hard with a single induced C4

Abstract

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 $C_4$, 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 $C_4$.