Research Article Counting Periodic Points in Parallel Graph Dynamical Systems

Hindawi Complexity Volume 2020, Article ID 9708347, 9 pages https://doi.org/10.1155/2020/9708347

Juan A. Aledo ,1 Ali Barzanouni ,2 Ghazaleh Malekbala ,2 Leila Sharifan ,2 and Jose C. Valverde 1

1Department of Mathematics, University of Castilla-La Mancha, Albacete, Spain 2Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran

Correspondence should be addressed to Jose C. Valverde; [email protected]

Received 27 March 2020; Accepted 22 May 2020; Published 14 September 2020

Academic Editor: Lucia Valentina Gambuzza

Copyright © 2020 Juan A. Aledo et al. 'is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let F: {}0, 1 n ⟶ {}0, 1 n 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 F 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 F. 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 graph G � (V, E) with vertex set V � {}1, ... , n and a Boolean function f: {}0, 1 n ⟶ {}0, 1 , we can define a ho- Dynamical systems have a long and brilliant history of mogeneous dynamical system, denoted by F: applications in biological networks [1], epidemic networks F: {}0, 1 n ⟶ {}0, 1 n, [2], social networks [3], and engineering control systems [4]. (1) In this paper, we deal with a particular class of finite dy- F x1, ... , xn � �y1, ... , yn �, namical systems, named deterministic Boolean networks [5, 6]. A deterministic Boolean network is a time-discrete such that each yi is given by the function fi which is the dynamical system F: {}0, 1 n ⟶ {}0, 1 n whose evolution restriction of the global function f to the state of the entity i operator is also a Boolean function. 'ese networks were and its neighbors. 'e graph G is called the dependency introduced by Kauffman [7] and also studied in works like graph of F. When all the entities are updated in a syn- [8–11] as models of genes activity and interactions. chronous manner, the system is called a parallel dynamical Besides the applications of Boolean networks in bio- system (PDS) [27–34], while if all the entities are updated in logical and social systems [12], these networks have been an asynchronous way, the system is called a sequential used in other branches of science for modeling problems in dynamical system [5, 6, 24, 35, 36]. When the function f is a different fields such as computer science [13, 14], chemistry maxterm (resp., a minterm), the system is called MAX − PDS [15, 16], mathematics [17, 18], and physics [19, 20]. In fact, (resp., MIN − PDS), in the parallel case, while in the se- due to their versatility, Boolean networks have become a quential case, it is named MAX-SDS (resp., MIN-SDS). profusely studied topic [5, 6, 21–26] in the last two decades. Here, by maxterm (resp., minterm), we mean Boolean networks can be represented by means of f(x1, ... , xn) � z1 ∨ ··· ∨ zn (resp., f(x1, ... , xn) � z1∧ ··· graphs. Specifically, associated with a (simple undirected) ∧zn), where zi � xi or zi � x′i. 2 Complexity

Following the notations of [33], we shall denote by points of an arbitrary MAX − PDS (and MIN − PDS), i.e., to W ⊆ V(resp., W′⊆V) the set of vertices in V such that the solve the minimum number problems. Besides, we provide corresponding variables in f appear in a direct (resp., formulae for counting (exactly) such points for PDS over complemented) form. Moreover, for each i ∈ V and Q ⊆ V, different particular classes of graphs. 'e results are shown we define here only for MAX − PDS, since they can be easily translated to the case of MIN − PDS by duality. Likewise, the results on AG(i) ��j: �i, j � ∈ E �, fixed points are also valid for MAX-SDS (and MIN-SDS) since, as well known, fixed points of an SDS are the same as A (i) � A (i) ∪ {}i , G G the corresponding PDS. (2) AG(Q) � ∪ AG(i), 'is paper is organized as follows. Section 2 is devoted to i Q ∈ counting fixed points and establishing the minimum AG(Q) � AG(Q) ∪ Q. number of them. For an arbitrary PDS on a maxterm, namely, F, over a dependency graph G � (V, E), we induce a _ Previous studies in deterministic Boolean (finite) net- bipartite graph BF on a vertex set V1 ∪ V2 (see Definition 2) works (also known as Boolean finite dynamical systems) and define the notion of dominating sets of V2 in BF (see show that the relationship between the structure of the Definition 1). Next, we provide a characterization of the network and its dynamics is a key issue. In this sense, some fixed points of F in terms of such dominating sets (see authors have set out the dynamical problem of finding fixed 'eorem 2). 'e bijective correspondence between the fixed points and periodic points (see, for instance, [26, 36–45]). points and dominating sets allows us to reformulate the From the point of view of complexity, the problem of finding known theorems on the existence and uniqueness of fixed fixed points (or, more generally, periodic points) of these points in terms of such dominating sets (see 'eorem 3). systems is NP-hard [46, 47]. Related to the problem of Such a correspondence also allows us to get the same upper counting periodic points, other problems such as the bound for number of fixed points as in [34]. Moreover, the maximum number problem and the minimum number new method to obtain this upper bound provides a char- problem are usually considered. In the literature, one can acterization to attain this maximum number of fixed points find some works on the study of the maximum number of (see Corollary 1). Finally, we consider minimal dominating fixed points of particular classes of Boolean finite dynamical sets and provide a lower (combinatorial) bound for the systems (see [37–39]). In particular, in [38, 39], the authors number of fixed points of F (see 'eorem 4). Furthermore, established a relationship between the maximum number of we are able to get a characterization for MAX − PDS to have fixed points in a special class of PDS and the maximum such minimum number of fixed points in 'eorem 5, thus number of maximal independent sets. In this paper, we use solving the minimum number of fixed-point problems for such maximal independent sets to establish the minimum this kind of PDS. Note that 'eorem 5 can be considered as number of 2-periodic points in homogeneous PDS induced a generalization of the fixed-point theorem ([33], 'eorem by a maxterm or minterm. Regarding the minimum number 9). In this section, we also provide a formula to easily count of fixed points’ problem, in the recent work [48], the authors (exactly) the fixed points (see 'eorem 6). In Section 3, we studied its complexity and established that finding a Boolean assume that F is a 2-periodic point MAX − PDS and study network having at least k fixed points is in P or complete for the number of its 2-periodic points. In this case, we use a NP, NP#P, or NEXPTIME, depending on the input. counting method based on the notion of maximal inde- In particular, for homogeneous PDS over a maxterm (or pendent sets of a graph and the corresponding indepen- minterm) Boolean function, a crucial result in [33] proves dence number. First, we provide a lower bound for the that if F is a MAX − PDS (or MIN − PDS) over a dependency number of 2-periodic points in the case of the maxterm graph G, then all the periodic points of this system are fixed NAND (see 'eorem 7). Moreover, in such a case, we points, or all of them are 2-periodic points. Actually, the provide a sufficient condition to attain this lower bound in authors found necessary and sufficient conditions under Corollary 3. As a consequence, we give a simple formula for which all the periodic points of a MAX − PDS (resp., a counting (exactly) the 2-periodic points when G is a t-partite MIN − PDS) are fixed points or all of them are 2-periodic graph or a (n − 2) regular graph (see Corollary 3). Finally, by points. When all the periodic points of the system are fixed combining the previous results, we give a lower combina- points, we simply say that F is a fixed point system and when torial bound for the number of 2-periodic points of a all of them are 2-periodic points, we say F is a 2-periodic MAX − PDS (see 'eorem 8). We also provide an example point system. In addition, a criterion was given to check if the to prove that this bound is the best possible one, i.e., it is the system has a unique fixed point. minimum number of 2-periodic points. We conclude the Later, in [34], the authors found a characterization of paper giving some conclusions and future research direc- MAX − PDS (resp., a MIN − PDS) to have a unique 2-pe- tions in Section 4. riodic orbit. Moreover, an upper bound for the number of fixed points and the number of 2-periodic points of such 2. Minimum Number of Fixed Points in PDS systems were also obtained in [34], thus solving the maximum number problems in this context. First, we recall a crucial theorem for characterizing In this work, our main goal is to provide lower bounds MAX − PDS which are fixed-point systems and for deter- for the number of fixed points and the number of periodic mining when a MAX − PDS has a unique fixed point. Complexity 3

Theorem 1 (see [33], 'eorems 3 and 9). Let F be a MAX − MAX − PDS, we define a bipartite graph BF in which _ PDS over a dependency graph G � (V, E), G1, ... ,Gp be the V(BF) � V1 ∪ V2 as connected components of G\W′, and (i) V1 � �1, ... , p �. Intuitively, we collapse the con- G i W1′ ��i ∈ W′: ∃!1 ≤ j ≤ p, such that i ∈ AG VG�j ��, nected component i into a single vertex for i � 1, ... , p. (3) (ii) V2 � W′. where V(Gj) is the set of vertices of Gj. ,en, (iii) 'e edge set of BF is defined as (1) F is a fixed point system if, and only if, W ∩ AG(i) ≠ ∅ for all i ∈ W′. (2) If F is a fixed point system, then F has a unique fixed �i, j � ∈ EBF � ⟺ ∃k ∈ Gi adjacent to j. (5) point if and only if for all 1 ≤ j ≤ p, AG(V(Gj)) ∩ W1′ ≠ ∅. In such a case, the unique fixed point of the system is the one with all the state values Example 1. Assume that G � (V, E) is the graph of Figure 1. equal to 1. Let us take

f x1, x2, x3, x4, x5, x6, x7, x8, x9� (6) � . Remark 1 (see [33], Section 3). We also know that if x � � x1 ∨ x2 ∨ x3 ∨ x4 ∨ x5 ∨ x6 ∨ x7′ ∨ x8′ ∨ x9′ (x1, ... , xn) is a fixed point, then 'en, W′ � {}7, 8, 9 and G\W′ has 4 connected com- W′ (fp1) All the variables associated with the vertices in ponents: G � {}1, 2 , which we collapse into the vertex 1; are activated 1 G2 � {}3 , which we collapse into the vertex 2; G3 � {}4 , i p (fp2) For each 1 ≤ ≤ , all the variables associated with which we collapse into the vertex 3; and G4 � {}5, 6 , which the vertices in Gi are activated, or all of them are we collapse into the vertex 4. 'en, bearing in mind the deactivated edges between the vertices in W′ and those in the com- (fp3) Moreover, if G ,G , ... ,G are the connected ponents Gi, the bipartite graph BF becomes as shown in j1 j2 jl components adjacent to a vertex j ∈ W′, then not all Figure 2: (the variables associated with the vertices in) these As a key point for our study, we then show that there is a components can be deactivated simultaneously; oth- bijective correspondence between the dominating sets of V2 B − erwise, the value of xj would change from 1 to 0 in the in F and the fixed points in the MAX PDS system. following iteration. F − From this characterization of fixed points, in 'eorem 1 Theorem 2. Let be a MAX PDS system over a depen- G � (V, E) W′ G , ... ,G of [34], it was stated that the number of fixed points of a dency graph with ≠ ∅, 1 p be the G\W′ B fixed-point system MAX − PDS over a connected graph with connected components of and F be the associated MAX ≠ OR is upper bounded by 2p − 1. For the well-known bipartite graph. ,en, there exists a bijective correspondence V B case, when MAX�OR, the number of fixed points is equal to between the dominating sets of 2 in F and the fixed points 2. of the system. Taking these results into account, we then provide a In particular novel combinatorial characterization of the fixed points of a � � − � � MAX PDS. In order to do this, we need some notions of |Fix(F)| � ��S ⊆ V1: S is a dominating set of V2 in BF��, graph theory. (7) Definition 1. Let B be a bipartite graph whose disjoint sets of where Fix(F) is the set of fixed points of the system. _ vertices are V1 and V2 (i.e., V1 ∪ V2 � V(B) and each edge of B has an endpoint in V and an endpoint in V ). 'en, we 1 2 Proof. Let Σ be the set of dominating sets of V2 in BF and let us say that S ⊆ V1 is a dominating set of V2 in B if every vertex in consider the map Ψ: Σ ⟶ Fix(F) defined as follows: given V is adjacent to at least one in S. In this situation, we define n 2 S ∈ Σ, we take Ψ(S) as the point x � (x1, ... , xn) ∈ {}0, 1 such that η(B) � min�|S|: S ⊆ V1 is a dominating set of V2 in B�.

(4) (Ψ1) All the variables associated with the vertices in W′ are activated Note that Definition 1 is an adaptation of the usual (Ψ ) For each i, 1 ≤ i ≤ p, if i ∈ S, then all the variables notion of dominating set in graph theory to bipartite graphs. 2 associated with the vertices in G are activated; oth- Next, for each MAX − PDS over a dependency graph G, i erwise, all the variables associated with the vertices in we induce a bipartite graph B. Gi are deactivated Definition 2. Let F be a MAX − PDS over a dependency Observe that Ψ(S) is a fixed point of the system (see graph G � (V, E) with ∅ ≠ W′⊊V, and let G1, ... ,Gp be the Remark 1), i.e., Ψ is well-defined. Moreover Ψ is injective connected components of G\W′. Associated with this since Ψ(S1) ≠ Ψ(S2) for S1,S2 ∈ Σ with S1 ≠ S2. Finally, given 4 Complexity

1 2 3 4 5 6 Corollary 1. Let F be a MAX − PDS system over a dependency graph G � (V, E) with ∅ ≠ W′⊊V, G1, ... ,Gp be the connected components of G\W′, and BF be the associated 7 8 9 p bipartite graph. ,en, |Fix(F)| � 2 − 1 if, and only if, BF is a Figure 1: Graph G. complete bipartite graph.

Proof. It is enough to note that, by 'eorem 2, |Fix(F)| is maximum if, and only if, each arbitrary nonempty subset of 1 2 3 4 V1 is a dominating set of V2 in BF which is equivalent to say that BF is a complete bipartite graph. □

7 8 9 Let us consider the order relation on Σ deﬁned by Figure S S S S ,S ,S . ( ) 2: Graph BF. 1 ≤ 2 ⟺ 1 ⊆ 2 1 2 ∈ Σ 8

Regarding this order, a dominating set S1 is said to be minimal if there is not another dominating set S such that a fixed point x ∈ Fix(F) and taking into account (fp1), (fp2), 2 S ⊈ S . and (fp3) in Remark 1, the subset S of V , whose elements are 2 1 1 Note that if S a dominating set of V and T ⊆ V is such the vertices i such that the vertices of G are activated in x, is 2 1 i that S ⊆ T, then T is also a dominating set. a dominating set of V verifying that Ψ(S) � x, i.e., Ψ is 2 With all of these, we can provide a lower bound for the surjective. □ number of fixed points in a PDS as follows. With all of these, we can rewrite 'eorem 1 to check if a F − MAX − PDS is a fixed-point system and if it has a unique Theorem 4. Let be a MAX PDS system over a dependency graph G � (V, E) with ∅ ≠ W′⊊V, G1, ... ,Gp be the fixed point in terms of the bipartite graph BF. connected components of G\W′, and BF be the associated bipartite graph. ,en, the number of fixed points is, at least, F − Theorem 3. Let be a MAX PDS system over a depen- 2p− η(BF). ,at is, dency graph G � (V, E) with ∅ ≠ W′⊊V, and BF, the associated bipartite graph. ,en, |Fix(F)| ≥ 2p− η()BF . (9)

(i) F is a fixed point system if, and only if, V1 is a dominating set of V2 in BF Proof Given S a dominating set such that |S| � η(BF), we (ii) F has a unique fixed point if, and only if, V is the 1 know that S ∪ T is also a dominating set of V2 for each unique dominating set of V in B 2 F T⊆V1\S. Since |V1\S| � p − η(BF), the possibilities of choosing T⊆V \S are 2p− η(BF). 'erefore, we have at least such Proof 1 number of dominating sets and, as a consequence of 'e- orem 2, at least such number of fixed points. □ (i) By 'eorem 1, F is a fixed point system if and only if W ∩ A (j) ≠ ∅ for all j ∈ W′. But this occurs, if and G Once we know the lower bound for the number of fixed only if, for all j ∈ W′, there exists i, 1 ≤ i ≤ p such that points, we can establish a minimum number of fixed-point j ∈ A (V(G )), which is equivalent to say that V is G i 1 theorem, thus generalizing the fixed-point theorem given for a dominating set of V in B . 'erefore, the con- 2 F PDS in [33], as follows. clusion follows. (ii) By 'eorem 2, every fixed point corresponds to a Theorem 5 (minimum number of fixed-point theorem). dominating set. 'us, we have a unique fixed point if, Let F be a MAX − PDS system over a dependency graph G � and only if, we have a unique dominating set in G. (V, E) with ∅ ≠ W′⊊V, G , ... ,G be the connected com- 'is occurs if, and only if, V is the unique domi- 1 p 1 ponents of G\W′, and BF be the associated bipartite graph. nating set. □ ,en, the number of fixed points is minimum if, and only if, there exists a unique minimal dominating set of V2 in BF. In Let F be a MAX − PDS over a dependency graph G � such a case, this minimum number of fixed points is given by (V, E) in which ∅ ≠ W′⊊V. In 'eorem 1 of [34], it is proved that the maximum number of fixed points for this |Fix(F)| � 2p− η()BF . (10) system is 2p − 1, where p is the number of connected components of G\W′. In the next corollary, we find a necessary and sufficient condition for achieving this upper Proof. Observe that the proof consists in getting the equality _ bound. Recall that a bipartite graph with V � V1 ∪ V2 is said in the formula of 'eorem 4 under the assumption of the to be a complete bipartite graph, if each node of V1 is adjacent existence of a unique minimal dominating set. Certainly, the to all the nodes in V2. equality follows under this assumption by taking into Complexity 5

account that, if S is the unique minimal dominating set of V2, 3 1 then |S| � η(BF) and every other dominating set of V2 is of the form S ∪ T for T⊆V1S. 2 4 5 Conversely, if S1 and S2 are two diﬀerent minimal dominating sets of V2 in BF and |S1| � η(BF), then S �S T; T V \S � F 2 ∉ 1 ∪ ⊆ 1 1 . 'us, would have at least 6 7 2p− η(BF) + 1 ﬁxed points. □ Figure 3: Graph G. Example 2. Let G be the graph of Figure 3. Assume that 1 2 3

f x1, x2, x3, x4, x5, x6, x7� � x1 ∨ x2 ∨ x3 ∨ x4 ∨ x5 ∨ x6′ ∨ x7′ . (11) 6 7 Figure 'en, by deﬁnition, BF is the graph of Figure 4. 4: Graph BF. It is easy to see that �1 � is the unique minimal dominating set of{} 6, 7 in B , η(B ) � 1, and G\W′ has 3 F F V B S � q S connected components. By 'eorem 4, F has 4 ﬁxed points minimal dominating sets of 2 in F, ∪ j�1 j, and as- which are (see 'eorem 3) sume that |S\Sj| � 1 for all j, 1 ≤ j ≤ q. ,en, Fix(F) �{ (1, 1, 1, 1, 1, 1, 1), (1, 1, 1, 1, 0, 1, 1), |Fix(F)| �(q + 1)2p− |S|. (17) (12) (1, 1, 1, 0, 0, 1, 1), (1, 1, 1, 0, 1, 1, 1)}.

Let us take M the set of minimal dominating sets of V2 Remark 2. Let F be an arbitrary MIN − PDS over a de- and pendency graph G � (V, E). 'en, all the results of this section can be restated for F by duality. S � ∪ S, (13) S∈M and then the union of all the minimal dominating sets of V2. 3. Minimum Number of 2-Periodic As stated in 'eorem 4, if |M| � 1, we have that Points in PDS

p− η()BF |Fix(F)| � 2 . (14) In this section, we provide a lower bound for the number of 2-periodic points of a MAX − PDS when it is a 2-periodic In this line, we are able to provide a formula for counting point system. To do this, we will use Proposition 3 in [34], (exactly) the fixed points of the system. where the number of 2-periodic orbits of a NAND-PDS is computed: Theorem 6. Let F be a MAX − PDS system over a dependency graph G � (V, E) with ∅ ≠ W′⊊V, G1, ... ,Gp be the Proposition 1 (see [34], Proposition 3). Let F be the connected components of G\W′, and BF be the associated NAND − PDS over the dependency graph G � (V, E) and bipartite graph. Let M � �S , ... ,Sq � be the set of minimal P(V) V 1 q be the power set of . ,en, the number of 2-periodic dominating sets of V2 in BF, S � ∪ j�1Sj, and points of this system is |Θ|, where � S V : S S S j j q . ( ) Ω � ⊆ 1 j ⊆ ⊆ for some 1 ≤ ≤ � 15 Θ � �AG(Q): Q ∈ P(V)�. (18) ,en, Specifically, in this proposition, it is proved that |Fix(F)| �|Ω|2p− |S|. (16) x �x1, ... , xn � ∈ Per2(F), if and only if �j: xj � 1 � ∈ Θ, (19) Proof. 'e equality follows from 'eorem 3 taking into where Per (F) is the set of 2-periodic points of the system. account that if S is a dominating set of V2, S ∪ T is also a 2 Given G � (V, E), we say that S ⊂ V is an independent dominating set of V2 for all T⊆V1\S. As a consequence, we have the following result when set of G if �i, j � ∉ E for whichever i, j ∈ S. Furthermore, S is q ≥ 2: □ said to be a maximal independent set if S ∪ {}k is not independent for all k ∈ V\S . 'e independence number of G, α(G), is then defined as Corollary 2. Let F be a MAX − PDS system over a depen- α(G) � max�|S|: S ⊆ V(G) is an independent set�. (20) dency graph G � (V, E) with ∅ ≠ W′⊊V, G1, ... ,Gp be the GW′ B connected components of , and F be the associated In the next result, we give a lower bound for the number bipartite graph. Let M � �S1, ... ,Sq � (q ≥ 2) be the set of of 2-periodic points of a NAND − PDS. 6 Complexity

2 Next, we compute the number of 2-periodic points of a NAND − PDS over a complete t-partite graph. Recall that the complete t-partite graph G � K is a graph whose set n1,...,nt 1 3 of nodes V is the union of pairwise disjoint subsets V1, ... ,Vt, with |Vi| � ni, and such that each vertex in Vi is adjacent to all the vertices in V\Vi, i � 1, ... , t. 6 4 Corollary 4. Let F be the NAND − PDS over the complete t-partite graph G � Kn ,...,n . ,en, 5 1 t � � t Figure 5: Graph G. � � ni �Per2(F)� � � 2 − 2(t − 1). (22) i�1 Theorem 7. Let F be the NAND − PDS over a dependency In particular, if G is the (n − 2)− regular graph, then graph G � (V, E). ,en, |Per (F)| � n + 2. � � 2 � � α(G) �Per2(F)� ≥ 2 . (21) Proof. Let G � Kn ,...,n be the complete t− partite graph on 1 t _ _ the vertex set V � V1 ∪ ··· ∪ Vt. For any ∅ ≠ Q⊆V, two Proof. Given F a NAND − PDS over G, we know by situations are possible: Proposition 1 that |Per2(F)| � |Θ|. Let S be a maximal independent set of G with |S| � α(G). (i) If Q⊆Vi for some 1 ≤ i ≤ t, then AG(Q) � Q ∪ (V\Vi) One can easily see that for any two distinct subsets Q1 and Q2 (ii) If Q is not contained in Vi for any 1 ≤ i ≤ t, then α(G) of S, AG(Q1) ≠ AG(Q2). 'erefore, Θ has, at least, 2 AG(Q) � V elements. □ Taking also into account that AG(∅) � ∅, we conclude that As a consequence, we have the following. Θ �{}V, ∅ ∪ �A (Q): ∅ ≠ Q⊊V , 1 ≤ i ≤ t�, (23) Corollary 3. Let F be the NAND − PDS over the dependency G i G � (V, E) S graph , and let be a maximal independent set of and so G such that |S| � α(G) and AG(i) � V for each i ∈ V\S. ,en, α(G) |Θ| � 2 . � � t � � ni �Per2(F)� �|Θ| � 2 + � 2 − 2t. (24) Proof. As shown in the proof of 'eorem 7, for every two i�1 different subsets of S, we have two different 2-periodic It is easy to check that if G is (n − 2)− regular, then n is points. even and G is a complete n/2-partite graph in which every V On the other hand, for any Q⊄S, there is i ∈ Q such that i has exactly 2elements (specifically each V consists in a pair i ∈ V\S. Hence, V � A (i) ⊂ A (Q). As S is a maximal i G G of nonadjacent vertices). 'erefore, independent set, A (S) � V � A (Q). 'us, only distinct G G � � subsets of S generate different periodic points, and the � � n 2 n �Per2(F)� � 2 + 2 − 2 � n + 2. (25) equality holds. □ 2 2 □

In particular, if G is a complete graph, then it satisﬁes the Given an arbitrary 2− periodic point system MAX − PDS conditions of the Corollary, being α(G) � 1, and the number over a dependency graph G � (V, E), we are able to obtain a of periodic points of the corresponding NAND − PDS co- | (F)| α(G) lower bound for Per2 by using 'eorems 4 and 7. We incides with 2 � 2 ([34], Proposition 1), i.e., the bound will assume that both W and W′ are nonempty sets since the provided in 'eorem 7 is attained. particular cases W � ∅ and W′ � ∅ have been already 'is situation in Corollary 3 also happens, for example, analyzed in detail. when G is a star graph where the center is the unique point Consider W′ as the disjoint union of WD′ and WC′, where outside the maximal independent set. 'us, the number of 2- n− 1 periodic points is 2 , which matches with the lower bound WD′ ��i ∈ W′: AG(i) ∩ W ≠ ∅�, ( ) (see [34], Example 2). 26 WC′ ��i ∈ W′: AG(i) ∩ W � ∅�. Let us present another example. 'e restriction of the system to the vertex set W ∪ WD′, Example 3. Let G be the graph of Figure 5. F| , is a ﬁxed-point MAX − PDS over the induced W∪WD′ One can easily see that{} 1, 2, 3 is a maximal independent subgraph GD of G whose vertex set is W ∪ WD′ (see [34], set in G and α(G) � 3. Moreover, AG(4) � Section 3). We also know that the restriction of F to WC′, A (5) � A (6) � {}1, 2, 3, 4, 5, 6 . 'en, from Corollary 3, we F| , is a 2− periodic point system over the induced sub- G G WC′ 3 have that |Θ| � 2 � 8. Actually, Θ � �AG(Q): Q ⊆ {}1, 2, 3 �. graph GC of G whose vertex set is WC′. Complexity 7

n Let x � (x1, ... , xn) be an arbitrary point of{} 0, 1 . 1 2 Without loss of generality, let us assume that x1, ... , xm are the variables associated with the vertices in W ∪ WD′ and xm+1, ... , xn those associated with the vertices in WC′. 3 4 5 Observe that if (x , ... , x ) is a fixed point of F| 1 m W∪WD′ and (x , ... , x ) is a 2− periodic point of F| , then x is a Figure 6: Graph G. m+1 n WC′ 2− periodic point of F. In this situation, the variables associated with the vertices in WD′ are fixed to 1, and so they do not affect the updating of the variables associated with 2 1 vertices in WC′ anymore (see [34], Section 3, for more details). 'erefore, 3 4 5 � � � � � � �Per (F)� ≥ �Per �F | �� ×�Fix�F | ��. (27) Figure 7: Graph G. 2 � 2 WC′ � � W∪WD′ � In fact, inequality (27) may be strict, as shown in the vertex set W ∪ W ′ � {}1, 2 ∪ {}3, 4 . For the corresponding following example. D bipartite graph BF| , we have GD � � VB�F | � � �1, 2 � ∪_ {}3, 4 , Example 4. Let G be the graph of Figure 6. GD Assume that F is the homogeneous MAX − PDS over G (32) � � � EB�F | � � ��1, 3 �, �1, 4 � � ∪ �2, 3 ��. whose evolution operator is demaxterm GD f x , x , x , x , x � � x x x′ x′ x′. ( ) 1 2 3 4 5 1 ∨ 2 ∨ 3 ∨ 4 ∨ 5 28 It is clear that �1 � is the unique minimal dominating set of V2 � {}3, 4 in BF| . 'us, by 'eorem 3, a point x � 'en, it can be easily checked that x � (0, 0, 1, 1, 0) is a GD (x , x , x , x , x ) is a fixed point of F| if and only if 2− periodic point of F (F(x) � (0, 0, 0, 1, 1)), but (0, 0, 1), 1 2 3 4 5 GD x � x � x � 1, and the number of fixed points of F| is associated with the vertices 1, 2, and 4, is not a fixed point of 1 3 4 GD 2− 1 G F| (F| (0, 0, 1) � (0, 0, 0)). Hence, for this system, in- 2 . Finally, note that C, the induced subgraph on the GD GD vertex set W ′, is an isolated vertex. So, Per (F | ) � equality (27) is strict. C 2 GC α(GC) {}(1), (0) and |Per (F | G )| � 2 � 2. By combining 'eorems 4 and 7, we have the following 2 C j j In the following, for each j ≥ 0, we denote by (x1, ... , x5) result: j 0 0 the point F (x) where x � (x1, ... , x5). 0 0 0 0 0 Assume that x � (x1, x2, x3, x4, x5) ∈ Per2(F). 'en, by Theorem 8. Let F be an arbitrary MAX − PDS over a de- 0 0 the same discussion before ([34], 'eorem 5), x3 � x4 � 1 pendency graph G with ∅ ≠ W′⊊V. If F is a 2− periodic point j j 0 (so, for each j ≥ 0, x3 � x4 � 1) and x5 � 0 or 1. system, then (x0, x0, x0, x0) F| We shall show that 1 2 3 4 is a fixed point of GD. 0 Actually, if it is not the case, then x1 � 0 and so, � � p− η B 0 0 � � �F | G � ( ) � � D α()GC 29 x � (0, x2, 1, 1, 1) or x � (0, x2, 1, 1, 0). Hence, looking to �Per2(F)� ≥ 2 2 , 1 2 f4 � x1 ∨ x4′ ∨ x5′ we have that x4 � 0 or x4 � 0 which is a (x0, x0, x0, x0) F| where p is the number of connected components of G\W′. contradiction. 'us, 1 2 3 4 is a fixed point of GD. 'erefore, a point x � (x1, x2, x3, x4, x5) is a 2− periodic We next give an example in which the number of point of F if, and only if, (x1, x2, x3, x4) is a fixed point of F| (x ) − F| 2− periodic points is exactly the same as the lower bound in GD and 5 is a 2 periodic point of GC. In particular, 'eorem 8. � � � � � � m− D � � � � � � η�F | G � � (F)� � � �F | �� ×� �F | �� D α()GC Per2 �Fix GD � �Per2 GC � 2 2 Example 5. Consider the graph G of Figure 7. 2− 1 2 � 2 × 2 � 2 . If we define f as (33) f x1, x2, x3, x4, x5� � x1 ∨ x2 ∨ x3′ ∨ x4′ ∨ x5′, (30) Specifically, in our case, Per2(F) �{ (1, 0, 1, 1, 1), and F is the corresponding homogeneous MAX − PDS, then (1, 0, 1, 1, 0), (1, 1, 1, 1, 1), (1, 1, 1, 1, 0)}. we have W �{}1, 2 , Remark 3. Let F be an arbitrary MIN − PDS over a dependency graph G � (V, E). 'en, all the results of this WD′ �{}3, 4 , (31) section can restated for F by duality. Moreover, since the

WC′ �{}5 . ﬁxed points of a PDS coincide with the ones of any SDS with the same dependency graph and local functions, whichever 'e two connected components of G\W′ � G\ the update order is, the results obtained in this section are (WD′ ∪ WC′) consist of the isolated vertices 1 and 2, re- also valid for the case of sequential dynamical systems of the spectively. Let GD be the induced subgraph of G on the form MAX-SDS and MIN-SDS. 8 Complexity

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