We are interested in fixed points in Boolean networks, { \em i.e.} functions $f$ from $\{ 0,1\}^n$ to itself. We define the subnetworks of $f$ as the restrictions of $f$ to the subcubes of $\{ 0,1\}^n$, and we characterizes a class $\mathcal{ F}$ of Boolean networks satisfying the following property: Every subnetwork of $f$ has a unique fixed point if and only if $f$ has no subnetwork in $\mathcal{ F}$. This characterization generalizes the fixed point theorem of Shih and Dong, which asserts that if for every $x$ in $\{ 0,1\}^n$ there is no directed cycle in the directed graph whose the adjacency matrix is the discrete Jacobian matrix of $f$ evaluated at point $x$, then $f$ has a unique fixed point. Then, denoting by $\mathcal{ C}^+$ (resp. $\mathcal{ C}^-$) the networks whose the interaction graph is a positive (resp. negative) cycle, we show that the non-expansive networks of $\mathcal{ F}$ are exactly the networks of $\mathcal{ C}^+\cup \mathcal{ C}^-$; and for the class of non-expansive networks we get a "dichotomization" of the previous forbidden subnetwork theorem: Every subnetwork of $f$ has at most (resp. at least) one fixed point if and only if $f$ has no subnetworks in $\mathcal{ C}^+$ (resp. $\mathcal{ C}^-$) subnetwork. Finally, we prove that if $f$ is a conjunctive network then every subnetwork of $f$ has at most one fixed point if and only if $f$ has no subnetwork in $\mathcal{ C}^+$.
Categories: cs.DM math.DS