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Let G be any graph and let {Hi}i∈I be a family of graphs such that E(Hi) ∩ E(Hj ) = ∅ when i 6= j, ∪i∈IE(Hi) = E(G) and E(Hi) 6= ∅ for all i ∈ I. In this paper we introduce the concept of {Hi}i∈I -super edge-magic decomposable graphs and {Hi}i∈I -super edge-magic labelings. We say that G is {Hi}i∈I -super edge-magic decomposable if there is a bijection β : V (G) → {1, 2, . . . , |V (G)|} such that for each i ∈ I the subgraph Hi meets the following two requirements: β(V (Hi)) = {1, 2, . . . , |V (Hi)|} and {β(a) + β(b) : ab ∈ E(Hi)} is a set of consecutive integers. Such function β is called an {Hi}i∈I -super edge-magic labeling of G. We characterize the set of cycles Cn which are {H1, H2}-super edge-magic decomposable when both, H1 and H2 are isomorphic to (n/2)K2. New lines of research are also suggested. The research conducted in this document by the first and third authors has been supported by the Spanish Research Council under project MTM2008- 06620-C03-01 and by the Catalan Research Council under grant 2009SGR1387.

At the beginning of 2005, Gheorghe Păun formulated a conjecture stating that in the framework of recognizer P systems with active membranes (evolution rules, communication rules, dissolution rules and division rules for elementary membranes), polarizations cannot be avoided in order to solve computationally hard problems efficiently (assuming that P ≠ NP ). At the middle of 2005, a partial positive answer was given, proving that the conjecture holds if dissolution rules are forbidden. In this paper we give a detailed and complete proof of this result modifying slightly the notion of dependency graph associated with recognizer P systems.

A tissue P system with cell division is a computing model which has two basic features: intercellular communication and the ability of cell division. The ability of cell division allows us to obtain an exponential amount of cells in linear time and to design cellular solutions to computationally hard problems in polynomial time. In this work we present an efficient solution to the tripartite matching problem by a family of such devices. This solution leads to an interesting open problem whether tissue P systems with cell division and communication rules of length 2 can solve NP-complete problems. An answer to this open problem will provide a borderline between efficiency and non-efficiency in terms of the lengths of communication rules Ministerio de Educación y Ciencia TIN2009-13192 Junta de Andalucía P08-TIC-04200

In 1936 A. Turing showed the existence of a universal machine able to simulate any Turing machine given its description. In 1956, C. Shannon formulated for the first time the problem of finding the smallest possible universal Turing machine according to some critera to measure its size such as the number of states and symbols. Within the framework of Membrane Computing different studies have addressed this problem: small universal symport/antiport P systems (by considering the number of membranes, the weight of the rules and the number of objects as a measure of the size of the system), small universal splicing P systems (by considering the number of rules as a measure of the size of the system), and small universal spiking neural P systems (by considering the number of neurons as a measure of the size of the system). In this paper the problem of determining the smallest possible efficient P system is explicitly formulated. Efficiency within the framework of Membrane Computing refers to the capability of solving computationally hard problems (i.e. problems such that classical electronic computer cannot solve instances of medium/large size in any reasonable amount of time) in polynomial time. A descriptive measure to define precisely the notion of small P system is presented in this paper.

Kotzig and Rosa defined in 1970 the concept of edge-magic labelings as follows: let G be a simple (p, q)-graph (that is, a graph of order p and size q without loops or multiple edges). A bijective function f : V (G)∪E(G) → {1, 2, . . . , p + q} is an edge-magic labeling of G if f(u) + f(uv) + f(v) = k, for all uv ∈ E(G). A graph that admits an edge-magic labeling is called an edgemagic graph, and k is called the magic sum of the labeling. An old conjecture of Godbold and Slater sets that all possible theoretical magic sums are attained for each cycle of order n ≥ 7. Motivated by this conjecture, we prove that for all n0 ∈ N, there exists n ∈ N, such that the cycle Cn admits at least n0 edge-magic labelings with at least n0 mutually distinct magic sums. We do this by providing a lower bound for the number of magic sums of the cycle Cn, depending on the sum of the exponents of the odd primes appearing in the prime factorization of n. The first and the third author are supported by the Spanish Research Council under project MTM2011-28800-C02-01 and by the Catalan Research Council under grant 2009SGR1387.

The efficiency of computational devices is usually expressed in terms of their capability to solve computationally hard problems in polynomial time. This paper focuses on tissue P systems, whose efficiency has been shown for several scenarios where the number of cells in the system can grow exponentially, e.g. by using cell division rules or cell separation rules. Moreover, in the first case it suffices to consider very short communication rules with length bounded by two, and in the second one it is enough to consider communication rules with length at most three. This kind of systems have an environment with the property that objects initially located in it appear in an arbitrarily large number of copies, which is a somewhat unfair condition from a computational complexity point of view. In this context, we study the role played by the environment and its ability to handle infinitely many objects, in particular we consider tissue P systems whose environment is initially empty. Ministerio de Ciencia e Innovación TIN2012-37434 Junta de Andalucía P08-TIC-04200

The standard definition of tissue P systems includes a special alphabet whose elements are assumed to appear in the initial configuration of the system in an arbitrarily large number of copies. These objects reside in a distinguished place of the system, called the environment. Such potentially infinite supply of objects seems an unfair tool when designing efficient solutions to computationally hard problems in the framework of membrane computing, by performing a space–time trade-off. This paper deals with computational aspects of tissue P systems with cell division where there is no environment having the property mentioned above. Specifically, we prove that the polynomial complexity classes associated with tissue P systems with cell division and with or without environment are actually identical. As a consequence, we conclude that it is not necessary to have infinitely many copies of some objects in the initial configuration in order to solve NP–complete problems in an efficient way. Ministerio de Ciencia e Innovación TIN2009–13192 Junta de Andalucía P08-TIC-04200

The classical definition of tissue P systems includes a distinguished alphabet with the special assumption that its elements are available in an arbitrarily large amount of copies. These objects are shared in a distinguished place of the system, called the environment. This ability of having infinitely many copies of some objects has been widely exploited in the design of efficient solutions to computationally hard problems by means of tissue P systems. This paper deals with computational aspects of tissue P systems with cell separation where there is no such environment as described above. The main result is that only tractable problems can be efficiently solved by using this kind of P systems. Bearing in mind that NP-complete problems can be efficiently solved by using tissue P systems without environment and with cell division, we deduce that in the framework of tissue P systems without environment, the kind of rules (separation versus division) provides a new frontier of the tractability of decision problems. Ministerio de Ciencia e Innovación TIN2009–13192 Junta de Andalucía P08-TIC-04200

A super edge-magic labeling of a graph G=(V,E) of order p and size q is a bijection f:V ∪E→{i}p+qi=1 such that: (1) f(u)+f(uv)+f(v)=k for all uv∈E; and (2) f(V )={i}pi=1. Furthermore, when G is a linear forest, the super edge-magic labeling of G is called strong if it has the extra property that if uv∈E(G) , u′,v′ ∈V (G) and dG (u,u′ )=dG (v,v′ )∞, then f(u)+f(v)=f(u′ )+f(v′ ). In this paper we introduce the concept of strong super edge-magic labeling of a graph G with respect to a linear forest F, and we study the super edge-magicness of an odd union of nonnecessarily isomorphic acyclic graphs. Furthermore, we find exponential lower bounds for the number of super edge-magic labelings of these unions. The case when G is not acyclic will be also considered.

In this paper, we generalize the concept of super edge-magic graph by introducing the new concept of super edge-magic models. The research conducted in this document by first and third author has been supported by the Spanish Research Council under project MTM2008-06620-C03-01 and by the Catalan Research Council under grant 2009SGR1387.

In this paper, we use the product ⊗h in order to study super edge-magic labelings, bi-magic labelings and optimal k-equitable labelings. We establish, with the help of the product ⊗h, new relations between super edge-magic labelings and optimal k-equitable labelings and between super edge-magic labelings and edge bi-magic labelings. We also introduce new families of graphs that are inspired by the family of generalized Petersen graphs. The concepts of super bi-magic and r-magic labelings are also introduced and discussed, and open problems are proposed for future research.

In the framework of cell–like membrane systems it is well known that the construction of exponential number of objects in polynomial time is not enough to efficiently solve NP–complete problems. Nonetheless, it may be sufficient to create an exponential number of membranes in polynomial time. In the framework of recognizer polarizationless P systems with active membranes, the construction of an exponential workspace expressed in terms of number of membranes and objects may not suffice to efficiently solve computationally hard problems. In this paper we study the computational efficiency of recognizer tissue P systems with communication (symport/antiport) rules and division rules. Some results have been already obtained in this direction: (a) using communication rules and forbidding division rules, only tractable problems can be efficiently solved; (b) using communication rules with length three and division rules, NP–complete problems can be efficiently solved. In this paper we show that the allowed length of communication rules plays a relevant role from the efficiency point of view of the systems. Ministerio de Educación y Ciencia TIN2006–13425 Junta de Andalucía P08-TIC-04200

A (p, q)-graph G is called super edge-magic if there exists a bijective function f : V (G) ∪ E(G) → {1, 2, . . . , p+q} such that f(u)+f(v)+f(uv) is a constant for each uv ∈ E(G) and f(V (G)) = {1, 2, . . . , p}. In this paper, we introduce the concept of strong super edge-magic labeling as a particular class of super edge-magic labelings and we use such labelings in order to show that the number of super edge-magic labelings of an odd union of path-like trees (mT), all of them of the same order, grows at least exponentially with m.

We study the structure of path-like trees. In order to do this, we introduce a set of trees that we call expandable trees. In this paper we also generalize the concept of path-like trees and we call such generalization generalized path-like trees. As in the case of path-like trees, generalized path-like trees, have very nice labeling properties.

The ⊗h-product was introduced in 2008 by Figueroa-Centeno et al. as a way to construct new families of (super) edge-magic graphs and to prove that some of those families admit an exponential number of (super) edge-magic labelings. In this paper, we extend the use of the product ⊗h in order to study the well know harmonious, sequential, partitional and (a, d)-edge antimagic total labelings. We prove that if a (p, q)-digraph with p ≤ q is harmonious and h : E(D) −→ Sn is any function, then und(D ⊗h Sn) is harmonious. We obtain analogous results for sequential and partitional labelings. We also prove that if G is a (super) (a, d)-edge-antimagic total tripartite graph, then nG is (super) (a ′ , d)-edge-antimagic total, where n ≥ 3, and d = 0, 2 and n is odd, or d = 1. We finish the paper providing an application of the product ⊗h to an arithmetic classical result when the function h is constant. The research conducted in this document by second and fourth author has been supported by the Spanish Research Council under project MTM2008-06620-C03-01 and by the Catalan Research Council under grant 2009SGR1387.

Let G be a graph of order p and size q with loops allowed. A bijective function f:V(G)∪E(G)→{i}p+qi=1 is an edge-magic labeling of G if the sum f(u)+f(uv)+f(v)=k is independent of the choice of the edge uv. The constant k is called either the valence, the magic weight or the magic sum of the labeling f. If a graph admits an edge-magic labeling, then it is called an edge-magic graph. Furthermore, if the function f meets the extra condition that f(V(G))={i}pi=1 then f is called a super edge-magic labeling and G is called a super edge-magic graph. A digraph D admits a labeling, namely l, if its underlying graph, und(D) admits l. In this paper, we introduce a new construction of super edge-magic labelings which are related to the classical jump of the knight on the chess game. We also use super edge-magic labelings of digraphs together with a generalization of the Kronecker product to get edge-magic labelings of some families of graphs. The research conducted in this document by the first and third authors has been supported by the Spanish Research Council under project MTM2011-28800-C02-01 and by the Catalan Research Council under grant 2009SGR1387.

It is well known that any irreducible and aperiodic Markov chain has exactly one stationary distribution, and for any arbitrary initial distribution, the se- quence of distributions at time n converges to the stationary distribution, that is, the Markov chain is approaching equilibrium as n→∞. In this paper, a characterization of the aperiodicity in existential terms of some state is given. At the same time, a P system with external output is associated with any irre- ducible Markov chain. The designed system provides the aperiodicity of that Markov chain and spends a polynomial amount of resources with respect to the size of the in- put. A comparative analysis with respect to another known solution is described. Ministerio de Educación y Ciencia TIN2006–13425 Junta de Andalucía P08-TIC-04200

In this paper we study the edge-magicness of graphs with equal size and order, and we use such graphs and digraph products in order to construct labelings of different classes and of different graphs. We also study super edge-magic labelings of 2-regular graphs with exactly two components and their implications to other labelings. The strength of the paper lays on the techniques used, since they are not only used in order to provide labelings of many different types of families of graphs, but they also show interesting relations among well studied types of labelings. We are able to obtain, in this way, deep results relating different types of labelings. The research conducted in this document by the first and third authors has been supported by the Spanish Research Council under project MTM2011-28800-C02-01 and by the Catalan Research Council under grant 2009SGR1387.