Miquel Rius-Font

BI-MAGIC AND OTHER GENERALIZATIONS OF SUPER EDGE-MAGIC LABELINGS

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.

The efficiency of tissue p systems with cell separation relies on the environment

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.

A polynomial alternative to unbounded environment for tissue P systems with cell division

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.

Labeling constructions using digraph products

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.

On the structure of paths-like trees

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.

ENUMERATING SUPER EDGE-MAGIC LABELINGS FOR THE UNION OF NONISOMORPHIC GRAPHS

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.

Super Edge-Magic Models

In this paper, we generalize the concept of super edge-magic graph by introducing the new concept of super edge-magic models.

ON A PARTIAL AFFIRMATIVE ANSWER FOR A PĂUN'S CONJECTURE

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.

The Relevance of the Environment on the Efficiency of Tissue P Systems

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.

Looking for Small Efficient P Systems

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.