#### Abstract

Let be a parallel dynamical system over an undirected graph with a Boolean maxterm or minterm function as a global evolution operator. It is well known that every periodic point has at most two periods. Actually, periodic points of different periods cannot coexist, and a fixed point theorem is also known. In addition, an upper bound for the number of periodic points of has been given. In this paper, we complete the study, solving the minimum number of periodic points’ problem for this kind of dynamical systems which has been usually considered from the point of view of complexity. In order to do this, we use methods based on the notions of minimal dominating sets and maximal independent sets in graphs, respectively. More specifically, we find a lower bound for the number of fixed points and a lower bound for the number of 2-periodic points of . In addition, we provide a formula that allows us to calculate the exact number of fixed points. Furthermore, we provide some conditions under which these lower bounds are attained, thus generalizing the fixed-point theorem and the 2-period theorem for these systems.

#### 1. Introduction

Dynamical systems have a long and brilliant history of applications in biological networks [1], epidemic networks [2], social networks [3], and engineering control systems [4]. In this paper, we deal with a particular class of finite dynamical systems, named *deterministic Boolean networks* [5, 6]. A deterministic Boolean network is a time-discrete dynamical system whose evolution operator is also a Boolean function. These networks were introduced by Kauffman [7] and also studied in works like [8–11] as models of genes activity and interactions.

Besides the applications of Boolean networks in biological and social systems [12], these networks have been used in other branches of science for modeling problems in different fields such as computer science [13, 14], chemistry [15, 16], mathematics [17, 18], and physics [19, 20]. In fact, due to their versatility, Boolean networks have become a profusely studied topic [5, 6, 21–26] in the last two decades.

Boolean networks can be represented by means of graphs. Specifically, associated with a (simple undirected) graph with vertex set and a Boolean function , we can define a *homogeneous* dynamical system, denoted by :such that each is given by the function which is the restriction of the global function to the state of the entity and its neighbors. The graph is called the *dependency graph* of . When all the entities are updated in a synchronous manner, the system is called a *parallel dynamical system* (PDS) [27–34], while if all the entities are updated in an asynchronous way, the system is called a sequential dynamical system [5, 6, 24, 35, 36]. When the function is a *maxterm* (resp., a *minterm*), the system is called (resp., ), in the parallel case, while in the sequential case, it is named MAX-SDS (resp., MIN-SDS). Here, by maxterm (resp., minterm), we mean , where or .

Following the notations of [33], we shall denote by (resp., ) the set of vertices in such that the corresponding variables in appear in a direct (resp., complemented) form. Moreover, for each and , we define

Previous studies in deterministic Boolean (finite) networks (also known as *Boolean finite dynamical systems*) show that the relationship between the structure of the network and its dynamics is a key issue. In this sense, some authors have set out the dynamical problem of finding fixed points and periodic points (see, for instance, [26, 36–45]). From the point of view of complexity, the problem of finding fixed points (or, more generally, periodic points) of these systems is NP-hard [46, 47]. Related to the problem of counting periodic points, other problems such as the maximum number problem and the minimum number problem are usually considered. In the literature, one can find some works on the study of the maximum number of fixed points of particular classes of Boolean finite dynamical systems (see [37–39]). In particular, in [38, 39], the authors established a relationship between the maximum number of fixed points in a special class of PDS and the maximum number of maximal independent sets. In this paper, we use such maximal independent sets to establish the minimum number of 2-periodic points in homogeneous PDS induced by a maxterm or minterm. Regarding the minimum number of fixed points’ problem, in the recent work [48], the authors studied its complexity and established that finding a Boolean network having at least fixed points is in *P* or complete for NP, P, or NEXPTIME, depending on the input.

In particular, for homogeneous PDS over a maxterm (or minterm) Boolean function, a crucial result in [33] proves that if is a (or ) over a dependency graph , then all the periodic points of this system are fixed points, or all of them are 2-periodic points. Actually, the authors found necessary and sufficient conditions under which all the periodic points of a (resp., a ) are fixed points or all of them are 2-periodic points. When all the periodic points of the system are fixed points, we simply say that is a *fixed point system* and when all of them are 2-periodic points, we say is a 2-*periodic point system*. In addition, a criterion was given to check if the system has a unique fixed point.

Later, in [34], the authors found a characterization of (resp., a ) to have a unique 2-periodic orbit. Moreover, an upper bound for the number of fixed points and the number of 2-periodic points of such systems were also obtained in [34], thus solving the maximum number problems in this context.

In this work, our main goal is to provide lower bounds for the number of fixed points and the number of periodic points of an arbitrary (and ), i.e., to solve the minimum number problems. Besides, we provide formulae for counting (exactly) such points for PDS over different particular classes of graphs. The results are shown here only for , since they can be easily translated to the case of by duality. Likewise, the results on fixed points are also valid for MAX-SDS (and MIN-SDS) since, as well known, fixed points of an SDS are the same as the corresponding PDS.

This paper is organized as follows. Section 2 is devoted to counting fixed points and establishing the minimum number of them. For an arbitrary PDS on a maxterm, namely, , over a dependency graph , we induce a bipartite graph on a vertex set (see Definition 2) and define the notion of dominating sets of in (see Definition 1). Next, we provide a characterization of the fixed points of in terms of such dominating sets (see Theorem 2). The bijective correspondence between the fixed points and dominating sets allows us to reformulate the known theorems on the existence and uniqueness of fixed points in terms of such dominating sets (see Theorem 3). Such a correspondence also allows us to get the same upper bound for number of fixed points as in [34]. Moreover, the new method to obtain this upper bound provides a characterization to attain this maximum number of fixed points (see Corollary 1). Finally, we consider minimal dominating sets and provide a lower (combinatorial) bound for the number of fixed points of (see Theorem 4). Furthermore, we are able to get a characterization for to have such minimum number of fixed points in Theorem 5, thus solving the minimum number of fixed-point problems for this kind of PDS. Note that Theorem 5 can be considered as a generalization of the fixed-point theorem ([33], Theorem 9). In this section, we also provide a formula to easily count (exactly) the fixed points (see Theorem 6). In Section 3, we assume that is a 2-periodic point and study the number of its 2-periodic points. In this case, we use a counting method based on the notion of maximal independent sets of a graph and the corresponding independence number. First, we provide a lower bound for the number of 2-periodic points in the case of the maxterm NAND (see Theorem 7). Moreover, in such a case, we provide a sufficient condition to attain this lower bound in Corollary 3. As a consequence, we give a simple formula for counting (exactly) the 2-periodic points when is a -partite graph or a regular graph (see Corollary 3). Finally, by combining the previous results, we give a lower combinatorial bound for the number of 2-periodic points of a (see Theorem 8). We also provide an example to prove that this bound is the best possible one, i.e., it is the minimum number of 2-periodic points. We conclude the paper giving some conclusions and future research directions in Section 4.

#### 2. Minimum Number of Fixed Points in PDS

First, we recall a crucial theorem for characterizing which are fixed-point systems and for determining when a has a unique fixed point.

Theorem 1 (see [33], Theorems 3 and 9). *Let be a over a dependency graph , be the connected components of , andwhere is the set of vertices of . Then,*(1)* is a fixed point system if, and only if, for all .*(2)*If is a fixed point system, then has a unique fixed point if and only if for all , . In such a case, the unique fixed point of the system is the one with all the state values equal to 1.*

*Remark 1. *(see [33], Section 3). We also know that if is a fixed point, then (fp1) All the variables associated with the vertices in are activated (fp2) For each , all the variables associated with the vertices in are activated, or all of them are deactivated (fp3) Moreover, if are the connected components adjacent to a vertex , then not all (the variables associated with the vertices in) these components can be deactivated simultaneously; otherwise, the value of would change from 1 to 0 in the following iteration.From this characterization of fixed points, in Theorem 1 of [34], it was stated that the number of fixed points of a fixed-point system over a connected graph with MAXOR is upper bounded by . For the well-known case, when MAXOR, the number of fixed points is equal to 2.

Taking these results into account, we then provide a novel combinatorial characterization of the fixed points of a . In order to do this, we need some notions of graph theory.

*Definition 1. *Let be a bipartite graph whose disjoint sets of vertices are and (i.e., and each edge of has an endpoint in and an endpoint in ). Then, we say that is a *dominating set of* in if every vertex in is adjacent to at least one in . In this situation, we defineNote that Definition 1 is an adaptation of the usual notion of dominating set in graph theory to bipartite graphs.

Next, for each over a dependency graph , we induce a bipartite graph .

*Definition 2. *Let be a over a dependency graph with , and let be the connected components of . Associated with this , we define a bipartite graph in which as(i). Intuitively, we collapse the connected component into a single vertex for .(ii).(iii)The edge set of is defined as

*Example 1. *Assume that is the graph of Figure 1.

Let us takeThen, and has 4 connected components: , which we collapse into the vertex ; , which we collapse into the vertex ; , which we collapse into the vertex ; and , which we collapse into the vertex . Then, bearing in mind the edges between the vertices in and those in the components , the bipartite graph becomes as shown in Figure 2:

As a key point for our study, we then show that there is a bijective correspondence between the dominating sets of in and the fixed points in the system.

Theorem 2. *Let be a system over a dependency graph with , be the connected components of and be the associated bipartite graph. Then, there exists a bijective correspondence between the dominating sets of in and the fixed points of the system.*

In particularwhere is the set of fixed points of the system.

*Proof. *Let be the set of dominating sets of in and let us consider the map defined as follows: given , we take as the point such that () All the variables associated with the vertices in are activated () For each , , if , then all the variables associated with the vertices in are activated; otherwise, all the variables associated with the vertices in are deactivatedObserve that is a fixed point of the system (see Remark 1), i.e., is well-defined. Moreover is injective since for with . Finally, given a fixed point and taking into account (fp1), (fp2), and (fp3) in Remark 1, the subset of , whose elements are the vertices such that the vertices of are activated in , is a dominating set of verifying that , i.e., is surjective.

With all of these, we can rewrite Theorem 1 to check if a is a fixed-point system and if it has a unique fixed point in terms of the bipartite graph .

Theorem 3. *Let be a system over a dependency graph with , and , the associated bipartite graph. Then,*(i)* is a fixed point system if, and only if, is a dominating set of in *(ii)* has a unique fixed point if, and only if, is the unique dominating set of in *

*Proof. *(i)By Theorem 1, is a fixed point system if and only if for all . But this occurs, if and only if, for all , there exists , such that , which is equivalent to say that is a dominating set of in . Therefore, the conclusion follows.(ii)By Theorem 2, every fixed point corresponds to a dominating set. Thus, we have a unique fixed point if, and only if, we have a unique dominating set in . This occurs if, and only if, is the unique dominating set.

Let be a over a dependency graph in which . In Theorem 1 of [34], it is proved that the maximum number of fixed points for this system is , where is the number of connected components of . In the next corollary, we find a necessary and sufficient condition for achieving this upper bound. Recall that a bipartite graph with is said to be a *complete bipartite graph*, if each node of is adjacent to all the nodes in .

Corollary 1. *Let be a system over a dependency graph with , be the connected components of , and be the associated bipartite graph. Then, if, and only if, is a complete bipartite graph.*

*Proof. *It is enough to note that, by Theorem 2, is maximum if, and only if, each arbitrary nonempty subset of is a dominating set of in which is equivalent to say that is a complete bipartite graph.

Let us consider the order relation on defined by

Regarding this order, a dominating set is said to be *minimal* if there is not another dominating set such that .

Note that if a dominating set of and is such that , then is also a dominating set.

With all of these, we can provide a lower bound for the number of fixed points in a PDS as follows.

Theorem 4. *Let be a system over a dependency graph with , be the connected components of , and be the associated bipartite graph. Then, the number of fixed points is, at least, . That is,*

*Proof. *Given a dominating set such that , we know that is also a dominating set of for each .

Since , the possibilities of choosing are . Therefore, we have at least such number of dominating sets and, as a consequence of Theorem 2, at least such number of fixed points.

Once we know the lower bound for the number of fixed points, we can establish a minimum number of fixed-point theorem, thus generalizing the fixed-point theorem given for PDS in [33], as follows.

Theorem 5 (minimum number of fixed-point theorem). * Let be a system over a dependency graph with , be the connected components of , and be the associated bipartite graph. Then, the number of fixed points is minimum if, and only if, there exists a unique minimal dominating set of in . In such a case, this minimum number of fixed points is given by*

*Proof. *Observe that the proof consists in getting the equality in the formula of Theorem 4 under the assumption of the existence of a unique minimal dominating set. Certainly, the equality follows under this assumption by taking into account that, if is the unique minimal dominating set of , then and every other dominating set of is of the form for .

Conversely, if are two different minimal dominating sets of in and , then . Thus, would have at least fixed points.

*Example 2. *Let be the graph of Figure 3.

Assume thatThen, by definition, is the graph of Figure 4.

It is easy to see that is the unique minimal dominating set of in , , and has 3 connected components. By Theorem 4, has 4 fixed points which are (see Theorem 3)Let us take the set of minimal dominating sets of andand then the union of all the minimal dominating sets of .

As stated in Theorem 4, if , we have thatIn this line, we are able to provide a formula for counting (exactly) the fixed points of the system.

Theorem 6. *Let be a system over a dependency graph with , be the connected components of , and be the associated bipartite graph. Let be the set of minimal dominating sets of in , , and**Then,*

*Proof. *The equality follows from Theorem 3 taking into account that if is a dominating set of , is also a dominating set of for all .

As a consequence, we have the following result when :

Corollary 2. *Let be a system over a dependency graph with , be the connected components of , and be the associated bipartite graph. Let () be the set of minimal dominating sets of in , , and assume that for all , . Then,*

*Remark 2. *Let be an arbitrary over a dependency graph . Then, all the results of this section can be restated for by duality.

#### 3. Minimum Number of 2-Periodic Points in PDS

In this section, we provide a lower bound for the number of 2-periodic points of a when it is a 2-periodic point system. To do this, we will use Proposition 3 in [34], where the number of 2-periodic orbits of a NAND-PDS is computed:

Proposition 1 (see [34], Proposition 3). *Let be the over the dependency graph and be the power set of . Then, the number of 2-periodic points of this system is , where*

Specifically, in this proposition, it is proved thatwhere is the set of 2-periodic points of the system.

Given , we say that is an independent set of if for whichever . Furthermore, is said to be a maximal independent set if is not independent for all . The independence number of , , is then defined as

In the next result, we give a lower bound for the number of 2-periodic points of a .

Theorem 7. *Let be the over a dependency graph . Then,*

*Proof. *Given a over , we know by Proposition 1 that .

Let be a maximal independent set of with . One can easily see that for any two distinct subsets and of , . Therefore, has, at least, elements.

As a consequence, we have the following.

Corollary 3. *Let be the over the dependency graph , and let be a maximal independent set of such that and for each . Then, .*

*Proof. *As shown in the proof of Theorem 7, for every two different subsets of , we have two different 2-periodic points.

On the other hand, for any , there is such that . Hence, . As is a maximal independent set, . Thus, only distinct subsets of generate different periodic points, and the equality holds.

In particular, if is a complete graph, then it satisfies the conditions of the Corollary, being , and the number of periodic points of the corresponding coincides with ([34], Proposition 1), i.e., the bound provided in Theorem 7 is attained.

This situation in Corollary 3 also happens, for example, when is a star graph where the center is the unique point outside the maximal independent set. Thus, the number of 2-periodic points is , which matches with the lower bound (see [34], Example 2).

Let us present another example.

*Example 3. *Let be the graph of Figure 5.

One can easily see that is a maximal independent set in and . Moreover, . Then, from Corollary 3, we have that . Actually, *.*

Next, we compute the number of 2-periodic points of a over a complete *t*-partite graph. Recall that the complete *t*-partite graph is a graph whose set of nodes is the union of pairwise disjoint subsets , with , and such that each vertex in is adjacent to all the vertices in , .

Corollary 4. *Let be the over the complete t-partite graph . Then,*

*In particular, if is the regular graph, then .*

*Proof. *Let be the complete partite graph on the vertex set . For any , two situations are possible:(i)If for some , then (ii)If is not contained in for any , then Taking also into account that , we conclude thatand soIt is easy to check that if is regular, then is even and is a complete -partite graph in which every has exactly elements (specifically each consists in a pair of nonadjacent vertices). Therefore,

Given an arbitrary periodic point system over a dependency graph , we are able to obtain a lower bound for by using Theorems 4 and 7. We will assume that both and are nonempty sets since the particular cases and have been already analyzed in detail.

Consider as the disjoint union of and , where

The restriction of the system to the vertex set , , is a fixed-point over the induced subgraph of whose vertex set is (see [34], Section 3). We also know that the restriction of to , , is a periodic point system over the induced subgraph of whose vertex set is .

Let be an arbitrary point of . Without loss of generality, let us assume that are the variables associated with the vertices in and those associated with the vertices in .

Observe that if is a fixed point of and is a periodic point of , then is a periodic point of . In this situation, the variables associated with the vertices in are fixed to 1, and so they do not affect the updating of the variables associated with vertices in anymore (see [34], Section 3, for more details).

Therefore,

In fact, inequality (27) may be strict, as shown in the following example.

*Example 4. *Let be the graph of Figure 6.

Assume that is the homogeneous over whose evolution operator is demaxtermThen, it can be easily checked that is a periodic point of (), but , associated with the vertices , and 4, is not a fixed point of (). Hence, for this system, inequality (27) is strict.

By combining Theorems 4 and 7, we have the following result:

Theorem 8. *Let be an arbitrary over a dependency graph with . If is a periodic point system, thenwhere is the number of connected components of .*

We next give an example in which the number of periodic points is exactly the same as the lower bound in Theorem 8.

*Example 5. *Consider the graph of Figure 7.

If we define asand is the corresponding homogeneous , then we haveThe two connected components of consist of the isolated vertices 1 and 2, respectively. Let be the induced subgraph of on the vertex set . For the corresponding bipartite graph , we haveIt is clear that is the unique minimal dominating set of in . Thus, by Theorem 3, a point is a fixed point of if and only if , and the number of fixed points of is . Finally, note that , the induced subgraph on the vertex set , is an isolated vertex. So, and .

In the following, for each , we denote by the point where .

Assume that . Then, by the same discussion before ([34], Theorem 5), (so, for each ) and or 1.

We shall show that is a fixed point of . Actually, if it is not the case, then and so, or . Hence, looking to we have that or which is a contradiction. Thus, is a fixed point of .

Therefore, a point is a periodic point of if, and only if, is a fixed point of and is a periodic point of . In particular*,*Specifically, in our case, .

*Remark 3. *Let be an arbitrary over a dependency graph . Then, all the results of this section can restated for by duality. Moreover, since the fixed points of a PDS coincide with the ones of any SDS with the same dependency graph and local functions, whichever the update order is, the results obtained in this section are also valid for the case of sequential dynamical systems of the form MAX-SDS and MIN-SDS.

#### 4. Conclusions and Future Research Directions

In this paper, we have shown that the problem of counting the fixed points of a homogeneous can be transformed into the problem of counting minimal dominating sets of an *ad hoc* bipartite graph constructed from the dependency graph of the system. The method used here has two advantages: first, we can get the results in previous works on the number of fixed points by means of more simple conditions in terms of minimal dominating sets; second, using this new approach, we have been able to provide a lower bound for the number of fixed points of an arbitrary and to generalize the fixed-point theorem stated in [33] with a minimum number of fixed point theorem; even more, we have also provided a formula to calculate the exact number of such fixed points. Of course, the results are also applicable to any MIN-PDS by duality. Moreover, since the fixed points of a PDS coincide with the ones of any SDS with the same dependency graph and local functions, whichever the update order is, the results on fixed points obtained are also valid for the case of sequential dynamical systems of the form MAX-SDS and MIN-SDS.

In addition, we have got a lower bound for the number of periodic points of an arbitrary and, in general, of any , in terms of the independence number of the dependency graph. Since the independence number is known for many classes of graphs (e.g., complete, path, cycle, and complete *k*-partite graphs), this method can be useful in many cases. By duality, such lower bounds are also valid to any NOR-PDS and any MIN-PDS, respectively.

Our results open some future research directions to be discovered. Actually, we think that, using ideas like the ones in this paper, periodic points of (resp., ) over directed graphs [27] could be better analyzed. The techniques employed here could be also interesting to extend this study to sequential dynamical systems, i.e., to those systems where the entities are updated asynchronously.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Authors’ Contributions

All authors contributed equally to this paper.

#### Acknowledgments

Juan A. Aledo was supported by Junta de Comunidades de Castilla-La Mancha under the Grant FEDER SBPLY/17/180501/000493. Jose C. Valverde was supported by FEDER OP2014-2020 of Castilla-La Mancha (Spain) under the Grant 2020-GRIN-29225 and by the Ministry of Science, Innovation and Universities of Spain under the Grant PGC2018-097198-B-I00.