Alejandro Estrada-Moreno | Universitat Rovira i Virgili (original) (raw)

Papers by Alejandro Estrada-Moreno

arXiv: Combinatorics, Oct 27, 2015

The General Randić index R α of a simple graph G is defined as R α (G) = v i ∼v j (d(v i)d(v j)) ... more The General Randić index R α of a simple graph G is defined as R α (G) = v i ∼v j (d(v i)d(v j)) α , where d(v i) denotes the degree of the vertex v i. Rodríguez-Velázquez and Tomás-Andreu [MATCH Commun. Math. Comput. Chem. 74 (1) (2015) 145-160] obtained closed formulae for the Randić index R −1/2 of Sierpiński-type polymeric networks, where the base graph is a complete graph, a triangle-free regular graph or a bipartite semiregular graph. In the present article we obtain closed formulae for the general Randić index R α of Sierpiński-type polymeric networks, where the base graph is arbitrary.

Bulletin of the Malaysian Mathematical Sciences Society, 2018

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension a... more The metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension and the local metric dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product G • H of a connected graph G of order n and a family H composed by n graphs. We show that the local metric dimension of G • H can be expressed in terms of the true twin equivalence classes of G and the local adjacency dimension of the graphs in H.

En aquesta tesi s'estudia la dimensio (k,t)-metrica dels grafs. Particularment s'emfatitz... more En aquesta tesi s'estudia la dimensio (k,t)-metrica dels grafs. Particularment s'emfatitza en la dimensio k-metrica i la dimensio de k-adjacencia. Al primer capitol es dedica als conceptes basics i les notacions emprades a la tesi. Al segon capitol es determina el mes gran enter k, de manera que hi ha una base (k,t) -metrica d'un graf. Es mostra, a mes, que la complexitat temporal de calcular aquest valor de k es cubica, pel que fa a l'ordre del graf. Al tercer capitol, s'obtenen formules tancades i cotes per a la dimensio (k,t) -metrica d'alguns grafs. Aixi mateix, es delimita el valor de la dimensio k-metrica en funcio de parametres relacionats amb la distancia, i es descriuen les classes de grafs en que s’aconsegueixen aquestes cotes com, per exemple, en els arbres. En particular, per a aquests ultims, s'estableix una formula per calcular la dimensio k-metrica de qualsevol arbre. Tambe s'estudia la dimensio de k-adjacencia de diversos grafs perque ...

Given a simple and connected graph G = (V, E), and a positive integer k, a set S ⊆ V is said to b... more Given a simple and connected graph G = (V, E), and a positive integer k, a set S ⊆ V is said to be a k-metric generator for G, if for any pair of different vertices u, v ∈ V , there exist at least k vertices w 1 , w 2 ,. .. , w k ∈ S such that d G (u, w i) = d G (v, w i), for every i ∈ {1,. .. , k}, where d G (x, y) denotes the distance between x and y. The minimum cardinality of a k-metric generator is the k-metric dimension of G. A set S ⊆ V is a k-adjacency generator for G if any two different vertices x, y ∈ V (G) satisfy |((N G (x)▽N G (y)) ∪ {x, y}) ∩ S| ≥ k, where N G (x)▽N G (y) is the symmetric difference of the neighborhoods of x and y. The minimum cardinality of any k-adjacency generator is the k-adjacency dimension of G. In this article we obtain tight bounds and closed formulae for the k-metric dimension of the lexicographic product of graphs in terms of the k-adjacency dimension of the factor graphs.

Ars Mathematica Contemporanea, 2022

In this paper, we show that the Italian domination number of every lexicographic product graph G ... more In this paper, we show that the Italian domination number of every lexicographic product graph G • H can be expressed in terms of five different domination parameters of G. These parameters can be defined under the following unified approach, which encompasses the definition of several well-known domination parameters and introduces new ones. Let N(v) denote the open neighbourhood of v ∈ V (G), and let w = (w 0 , w 1 ,. .. , w l) be a vector of nonnegative integers such that w 0 ≥ 1. We say that a function f : V (G) −→ {0, 1,. .. , l} is a w-dominating function if f (N(v)) = ∑ u∈N(v) f (u) ≥ w i for every vertex v with f (v) = i. The weight of f is defined to be ω(f) = ∑ v∈V (G) f (v). The w-domination number of G, denoted by γ w (G), is the minimum weight among all w-dominating functions on G. Specifically, we show that γ I (G • H) = γ w (G), where w ∈ {2} × {0, 1, 2} l and l ∈ {2, 3}. The decision on whether the equality holds for specific values of w 0 ,. .. , w l will depend on the value of the domination number of H. This paper also provides preliminary results on γ w (G) and raises the challenge of conducting a detailed study of the topic.

2019 Winter Simulation Conference (WSC), 2019

The social and economic impact of natural catastrophes on communities is a concern for many gover... more The social and economic impact of natural catastrophes on communities is a concern for many governments and corporations across the globe. A class of financial instruments, parametric hedges, is emerging in the (re)insurance market as a promising approach to close the protection gap related to natural hazards. This paper focuses on the design of such parametric hedges, which have the objective of maximizing the risk transferred subject to a budget constraint. With Greece as a case study, one of the most seismic prone European regions, with limited seismic insurance penetration, this paper proposes a biased-randomized algorithm to solve the optimization problem. The algorithm hybridizes Monte Carlo simulation with a heuristic to generate a variety of solutions. A simulation stage allows for analyzing the payout distribution of each solution. Results show the impact of the problem resolution on the transferred risk and on the distribution of triggered payments.

Symmetry, 2021

This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a... more This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex x∈V(G) of a graph G, the neighbourhood of x is denoted by N(x). The neighbourhood of a set X⊆V(G) is defined to be N(X)=⋃x∈XN(x), while the external neighbourhood of X is defined to be Ne(X)=N(X)∖X. Now, for every set X⊆V(G) and every vertex x∈X, the external private neighbourhood of x with respect to X is defined as the set Pe(x,X)={y∈V(G)∖X:N(y)∩X={x}}. Let Xw={x∈X:Pe(x,X)≠⌀}. The strong differential of X is defined to be ∂s(X)=|Ne(X)|−|Xw|, while the quasi-total strong differential of G is defined to be ∂s*(G)=max{∂s(X):X⊆V(G)andXw⊆N(X)}. We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination...

Diversos autores consideran que la eclosion de Internet en las campanas electorales ha cambiado l... more Diversos autores consideran que la eclosion de Internet en las campanas electorales ha cambiado las estrategias de los partidos politicos para presentar sus propuestas de gobierno, por el potencial uso del medio para debatir o interactuar interpoliticos y ciudadanos, otros ven a la web como una nueva vitrina para los programas ideopoliticos. En este contexto, se pretende explicar si la campana electoral en Twitter del PSOE y de Podemos, previa a las elecciones locales y autonomicas de 2015, aprovecha las potencialidades de interaccion de la comunicacion politica online. Tal objetivo aplica parte del diseno teorico-metodologico de una investigacion doctoral. Asimismo expone el funcionamiento de dos programas informaticos: uno creado para la captura y almacenamiento de tuits; y otro disenado para procesar cuantitativamente los datos. Los resultados, apoyados en metodos cuantitativos y cualitativos, contribuyen a establecer hipotesis sobre las estrategias de campana en Twitter durante ...

Given a connected graph G=(V,E)G=(V,E)G=(V,E), a set SsubseteqVS\subseteq VSsubseteqV is a kkk-metric generator for GGG if for ... more Given a connected graph G=(V,E)G=(V,E)G=(V,E), a set SsubseteqVS\subseteq VSsubseteqV is a kkk-metric generator for GGG if for any two different vertices u,vinVu,v\in Vu,vinV, there exist at least kkk vertices w1,...,wkinSw_1,...,w_k\in Sw1,...,wkinS such that dG(u,wi)nedG(v,wi)d_G(u,w_i)\ne d_G(v,w_i)dG(u,wi)nedG(v,wi) for every iin1,...,ki\in \{1,...,k\}iin1,...,k. A metric generator of minimum cardinality is called a kkk-metric basis and its cardinality the kkk-metric dimension of GGG. We study some problems regarding the complexity of some kkk-metric dimension problems. For instance, we show that the problem of computing the kkk-metric dimension of graphs is NPNPNP-Complete. However, the problem is solved in linear time for the particular case of trees.

Mathematics, 2021

Given a connected graph G=(V(G),E(G)), a set S⊆V(G) is said to be a k-metric generator for G if a... more Given a connected graph G=(V(G),E(G)), a set S⊆V(G) is said to be a k-metric generator for G if any pair of different vertices in V(G) is distinguished by at least k elements of S. A metric generator of minimum cardinality among all k-metric generators is called a k-metric basis and its cardinality is the k-metric dimension of G. We initially present a linear programming problem that describes the problem of finding the k-metric dimension and a k-metric basis of a graph G. Then we conducted a study on the k-metric dimension of a unicyclic graph.

Symmetry, 2020

This paper introduces a general approach to the idea of protection of graphs, which encompasses t... more This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure w-domination in graphs, where w=(w0,w1,…,wl) is a vector of nonnegative integers such that w0≥1. The secure w-domination number is defined as follows. Let G be a graph and N(v) the open neighborhood of v∈V(G). We say that a function f:V(G)⟶{0,1,…,l} is a w-dominating function if f(N(v))=∑u∈N(v)f(u)≥wi for every vertex v with f(v)=i. The weight of f is defined to be ω(f)=∑v∈V(G)f(v). Given a w-dominating function f and any pair of adjacent vertices v,u∈V(G) with f(v)=0 and f(u)>0, the function fu→v is defined by fu→v(v)=1, fu→v(u)=f(u)−1 and fu→v(x)=f(x) for every x∈V(G)\{u,v}. We say that a w-dominating function f is a secure w-dominating function if for every v with f(v)=0, there exists u∈N(v) such that f(u)>0 and fu→v is a w-dominating function as well. The secure w-domi...

Bulletin of the Malaysian Mathematical Sciences Society, 2021

Let w = (w 0 , w 1 ,. .. , w l) be a vector of nonnegative integers such that w 0 ≥ 1. Let G be a... more Let w = (w 0 , w 1 ,. .. , w l) be a vector of nonnegative integers such that w 0 ≥ 1. Let G be a graph and N(v) the open neighbourhood of v ∈ V (G). We say that a function f : V (G) −→ {0, 1,. .. , l} is a w-dominating function if f (N(v)) = ∑ u∈N(v) f (u) ≥ w i for every vertex v with f (v) = i. The weight of f is defined to be ω(f) = ∑ v∈V (G) f (v). Given a w-dominating function f and any pair of adjacent vertices v, u ∈ V (G) with f (v) = 0 and f (u) > 0, the function f u→v is defined by f u→v (v) = 1, f u→v (u) = f (u) − 1 and f u→v (x) = f (x) for every x ∈ V (G) \ {u, v}. We say that a w-dominating function f is a secure w-dominating function if for every v with f (v) = 0, there exists u ∈ N(v) such that f (u) > 0 and f u→v is a w-dominating function as well. The (secure) w-domination number of G, denoted by (γ s w (G)) γ w (G), is defined as the minimum weight among all (secure) w-dominating functions. In this paper, we show how the secure (total) domination number and the (total) weak Roman domination number of lexicographic product graphs G • H are related to γ s w (G) or γ w (G). For the case of the secure domination number and the weak Roman domination number, the decision on whether w takes specific components will depend on the value of γ s (1,0) (H), while in the case of the total version of these parameters, the decision will depend on the value of γ s (1,1) (H).

Mathematics, Apr 15, 2020

In this article, we obtain general bounds and closed formulas for the secure total domination num... more In this article, we obtain general bounds and closed formulas for the secure total domination number of rooted product graphs. The results are expressed in terms of parameters of the factor graphs involved in the rooted product.

Applicable Analysis and Discrete Mathematics, 2018

In this paper we propose formulas for the distance between vertices of a generalized Sierpi?ski g... more In this paper we propose formulas for the distance between vertices of a generalized Sierpi?ski graph S(G, t) in terms of the distance between vertices of the base graph G. In particular, we deduce a recursive formula for the distance between an arbitrary vertex and an extreme vertex of S(G, t), and we obtain a recursive formula for the distance between two arbitrary vertices of S(G, t) when the base graph is triangle-free. From these recursive formulas, we provide algorithms to compute the distance between vertices of S(G, t). In addition, we give an explicit formula for the diameter and radius of S(G, t) when the base graph is a tree.

The Computer Journal, 2020

Let (X,d)(X,d)(X,d) be a metric space. A set SsubseteqXS\subseteq XSsubseteqX is said to be a kkk-metric generator for XXX ... more Let (X,d)(X,d)(X,d) be a metric space. A set SsubseteqXS\subseteq XSsubseteqX is said to be a kkk-metric generator for XXX if and only if for any pair of different points u,vinXu,v\in Xu,vinX, there exist at least kkk points w1,w2,ldotswkinSw_1,w_2, \ldots w_k\in Sw1,w2,ldotswkinS such that d(u,wi)ned(v,wi),;textrmforall;iin1,ldotsk.d(u,w_i)\ne d(v,w_i),\; \textrm{for all}\; i\in \{1, \ldots k\}.d(u,wi)ned(v,wi),;textrmforall;iin1,ldotsk. Let mathcalRk(X)\mathcal{R}_k(X)mathcalRk(X) be the set of metric generators for XXX. The kkk-metric dimension dimk(X)\dim _k(X)dimk(X) of (X,d)(X,d)(X,d) is defined as \begin{equation*}\dim_k(X)=\inf\{|S|:\, S\in \mathcal{R}_k(X)\}.\end{equation*}$$Here, we discuss the kkk-metric dimension of (V,dt)(V,d_t)(V,dt), where VVV is the set of vertices of a simple graph GGG and the metric dt:VtimesVrightarrowmathbbNcup0d_t:V\times V\rightarrow \mathbb{N}\cup \{0\}dt:VtimesVrightarrowmathbbNcup0 is defined by dt(x,y)=mind(x,y),td_t(x,y)=\min \{d(x,y),t\}dt(x,y)=mind(x,y),t from the geodesic distance ddd in GGG and a positive integer ttt. The case tgeD(G)t\ge D(G)tgeD(G), where D(G)D(G)D(G) denotes the diameter of GGG, corresponds to the original theory of kkk-metric dimension, and the case t=2t=2t=2 corresponds to the theory of kkk-adjacency dimension. Furthermore, t...

International Transactions in Operational Research, 2020

Theoretical Computer Science, 2020

As a generalization of the concept of the partition dimension of a graph, this article introduces... more As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the k-partition dimension. Given a nontrivial connected graph G = (V, E), a partition Π of V is said to be a k-partition generator of G if any pair of different vertices u, v ∈ V is distinguished by at least k vertex sets of Π, i.e., there exist at least k vertex sets S 1 ,. .. , S k ∈ Π such that d(u, S i) = d(v, S i) for every i ∈ {1,. .. , k}. A k-partition generator of G with minimum cardinality among all their k-partition generators is called a k-partition basis of G and its cardinality the k-partition dimension of G. A nontrivial connected graph G is k-partition dimensional if k is the largest integer such that G has a k-partition basis. We give a necessary and sufficient condition for a graph to be r-partition dimensional and we obtain several results on the k-partition dimension for k ∈ {1,. .. , r}.

Journal of the Operational Research Society, 2019

Discrete Applied Mathematics, 2019

The General Randić index R α of a simple graph G is defined as R α (G) = v i ∼v j (d(v i)d(v j)) ... more The General Randić index R α of a simple graph G is defined as R α (G) = v i ∼v j (d(v i)d(v j)) α , where d(v i) denotes the degree of the vertex v i. Rodríguez-Velázquez and Tomás-Andreu [MATCH Commun. Math. Comput. Chem. 74 (1) (2015) 145-160] obtained closed formulae for the Randić index R −1/2 of Sierpiński-type polymeric networks, where the base graph is a complete graph, a triangle-free regular graph or a bipartite semiregular graph. In the present article we obtain closed formulae for the general Randić index R α of Sierpiński-type polymeric networks, where the base graph is arbitrary.

arXiv: Combinatorics, Oct 27, 2015

The General Randić index R α of a simple graph G is defined as R α (G) = v i ∼v j (d(v i)d(v j)) ... more The General Randić index R α of a simple graph G is defined as R α (G) = v i ∼v j (d(v i)d(v j)) α , where d(v i) denotes the degree of the vertex v i. Rodríguez-Velázquez and Tomás-Andreu [MATCH Commun. Math. Comput. Chem. 74 (1) (2015) 145-160] obtained closed formulae for the Randić index R −1/2 of Sierpiński-type polymeric networks, where the base graph is a complete graph, a triangle-free regular graph or a bipartite semiregular graph. In the present article we obtain closed formulae for the general Randić index R α of Sierpiński-type polymeric networks, where the base graph is arbitrary.

Bulletin of the Malaysian Mathematical Sciences Society, 2018

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension a... more The metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension and the local metric dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product G • H of a connected graph G of order n and a family H composed by n graphs. We show that the local metric dimension of G • H can be expressed in terms of the true twin equivalence classes of G and the local adjacency dimension of the graphs in H.

En aquesta tesi s'estudia la dimensio (k,t)-metrica dels grafs. Particularment s'emfatitz... more En aquesta tesi s'estudia la dimensio (k,t)-metrica dels grafs. Particularment s'emfatitza en la dimensio k-metrica i la dimensio de k-adjacencia. Al primer capitol es dedica als conceptes basics i les notacions emprades a la tesi. Al segon capitol es determina el mes gran enter k, de manera que hi ha una base (k,t) -metrica d'un graf. Es mostra, a mes, que la complexitat temporal de calcular aquest valor de k es cubica, pel que fa a l'ordre del graf. Al tercer capitol, s'obtenen formules tancades i cotes per a la dimensio (k,t) -metrica d'alguns grafs. Aixi mateix, es delimita el valor de la dimensio k-metrica en funcio de parametres relacionats amb la distancia, i es descriuen les classes de grafs en que s’aconsegueixen aquestes cotes com, per exemple, en els arbres. En particular, per a aquests ultims, s'estableix una formula per calcular la dimensio k-metrica de qualsevol arbre. Tambe s'estudia la dimensio de k-adjacencia de diversos grafs perque ...

Given a simple and connected graph G = (V, E), and a positive integer k, a set S ⊆ V is said to b... more Given a simple and connected graph G = (V, E), and a positive integer k, a set S ⊆ V is said to be a k-metric generator for G, if for any pair of different vertices u, v ∈ V , there exist at least k vertices w 1 , w 2 ,. .. , w k ∈ S such that d G (u, w i) = d G (v, w i), for every i ∈ {1,. .. , k}, where d G (x, y) denotes the distance between x and y. The minimum cardinality of a k-metric generator is the k-metric dimension of G. A set S ⊆ V is a k-adjacency generator for G if any two different vertices x, y ∈ V (G) satisfy |((N G (x)▽N G (y)) ∪ {x, y}) ∩ S| ≥ k, where N G (x)▽N G (y) is the symmetric difference of the neighborhoods of x and y. The minimum cardinality of any k-adjacency generator is the k-adjacency dimension of G. In this article we obtain tight bounds and closed formulae for the k-metric dimension of the lexicographic product of graphs in terms of the k-adjacency dimension of the factor graphs.

Ars Mathematica Contemporanea, 2022

In this paper, we show that the Italian domination number of every lexicographic product graph G ... more In this paper, we show that the Italian domination number of every lexicographic product graph G • H can be expressed in terms of five different domination parameters of G. These parameters can be defined under the following unified approach, which encompasses the definition of several well-known domination parameters and introduces new ones. Let N(v) denote the open neighbourhood of v ∈ V (G), and let w = (w 0 , w 1 ,. .. , w l) be a vector of nonnegative integers such that w 0 ≥ 1. We say that a function f : V (G) −→ {0, 1,. .. , l} is a w-dominating function if f (N(v)) = ∑ u∈N(v) f (u) ≥ w i for every vertex v with f (v) = i. The weight of f is defined to be ω(f) = ∑ v∈V (G) f (v). The w-domination number of G, denoted by γ w (G), is the minimum weight among all w-dominating functions on G. Specifically, we show that γ I (G • H) = γ w (G), where w ∈ {2} × {0, 1, 2} l and l ∈ {2, 3}. The decision on whether the equality holds for specific values of w 0 ,. .. , w l will depend on the value of the domination number of H. This paper also provides preliminary results on γ w (G) and raises the challenge of conducting a detailed study of the topic.

2019 Winter Simulation Conference (WSC), 2019

The social and economic impact of natural catastrophes on communities is a concern for many gover... more The social and economic impact of natural catastrophes on communities is a concern for many governments and corporations across the globe. A class of financial instruments, parametric hedges, is emerging in the (re)insurance market as a promising approach to close the protection gap related to natural hazards. This paper focuses on the design of such parametric hedges, which have the objective of maximizing the risk transferred subject to a budget constraint. With Greece as a case study, one of the most seismic prone European regions, with limited seismic insurance penetration, this paper proposes a biased-randomized algorithm to solve the optimization problem. The algorithm hybridizes Monte Carlo simulation with a heuristic to generate a variety of solutions. A simulation stage allows for analyzing the payout distribution of each solution. Results show the impact of the problem resolution on the transferred risk and on the distribution of triggered payments.

Symmetry, 2021

This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a... more This paper is devoted to the study of the quasi-total strong differential of a graph, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. Given a vertex x∈V(G) of a graph G, the neighbourhood of x is denoted by N(x). The neighbourhood of a set X⊆V(G) is defined to be N(X)=⋃x∈XN(x), while the external neighbourhood of X is defined to be Ne(X)=N(X)∖X. Now, for every set X⊆V(G) and every vertex x∈X, the external private neighbourhood of x with respect to X is defined as the set Pe(x,X)={y∈V(G)∖X:N(y)∩X={x}}. Let Xw={x∈X:Pe(x,X)≠⌀}. The strong differential of X is defined to be ∂s(X)=|Ne(X)|−|Xw|, while the quasi-total strong differential of G is defined to be ∂s*(G)=max{∂s(X):X⊆V(G)andXw⊆N(X)}. We show that the quasi-total strong differential is closely related to several graph parameters, including the domination number, the total domination number, the 2-domination number, the vertex cover number, the semitotal domination...

Diversos autores consideran que la eclosion de Internet en las campanas electorales ha cambiado l... more Diversos autores consideran que la eclosion de Internet en las campanas electorales ha cambiado las estrategias de los partidos politicos para presentar sus propuestas de gobierno, por el potencial uso del medio para debatir o interactuar interpoliticos y ciudadanos, otros ven a la web como una nueva vitrina para los programas ideopoliticos. En este contexto, se pretende explicar si la campana electoral en Twitter del PSOE y de Podemos, previa a las elecciones locales y autonomicas de 2015, aprovecha las potencialidades de interaccion de la comunicacion politica online. Tal objetivo aplica parte del diseno teorico-metodologico de una investigacion doctoral. Asimismo expone el funcionamiento de dos programas informaticos: uno creado para la captura y almacenamiento de tuits; y otro disenado para procesar cuantitativamente los datos. Los resultados, apoyados en metodos cuantitativos y cualitativos, contribuyen a establecer hipotesis sobre las estrategias de campana en Twitter durante ...

Given a connected graph G=(V,E)G=(V,E)G=(V,E), a set SsubseteqVS\subseteq VSsubseteqV is a kkk-metric generator for GGG if for ... more Given a connected graph G=(V,E)G=(V,E)G=(V,E), a set SsubseteqVS\subseteq VSsubseteqV is a kkk-metric generator for GGG if for any two different vertices u,vinVu,v\in Vu,vinV, there exist at least kkk vertices w1,...,wkinSw_1,...,w_k\in Sw1,...,wkinS such that dG(u,wi)nedG(v,wi)d_G(u,w_i)\ne d_G(v,w_i)dG(u,wi)nedG(v,wi) for every iin1,...,ki\in \{1,...,k\}iin1,...,k. A metric generator of minimum cardinality is called a kkk-metric basis and its cardinality the kkk-metric dimension of GGG. We study some problems regarding the complexity of some kkk-metric dimension problems. For instance, we show that the problem of computing the kkk-metric dimension of graphs is NPNPNP-Complete. However, the problem is solved in linear time for the particular case of trees.

Mathematics, 2021

Given a connected graph G=(V(G),E(G)), a set S⊆V(G) is said to be a k-metric generator for G if a... more Given a connected graph G=(V(G),E(G)), a set S⊆V(G) is said to be a k-metric generator for G if any pair of different vertices in V(G) is distinguished by at least k elements of S. A metric generator of minimum cardinality among all k-metric generators is called a k-metric basis and its cardinality is the k-metric dimension of G. We initially present a linear programming problem that describes the problem of finding the k-metric dimension and a k-metric basis of a graph G. Then we conducted a study on the k-metric dimension of a unicyclic graph.

Symmetry, 2020

This paper introduces a general approach to the idea of protection of graphs, which encompasses t... more This paper introduces a general approach to the idea of protection of graphs, which encompasses the known variants of secure domination and introduces new ones. Specifically, we introduce the study of secure w-domination in graphs, where w=(w0,w1,…,wl) is a vector of nonnegative integers such that w0≥1. The secure w-domination number is defined as follows. Let G be a graph and N(v) the open neighborhood of v∈V(G). We say that a function f:V(G)⟶{0,1,…,l} is a w-dominating function if f(N(v))=∑u∈N(v)f(u)≥wi for every vertex v with f(v)=i. The weight of f is defined to be ω(f)=∑v∈V(G)f(v). Given a w-dominating function f and any pair of adjacent vertices v,u∈V(G) with f(v)=0 and f(u)>0, the function fu→v is defined by fu→v(v)=1, fu→v(u)=f(u)−1 and fu→v(x)=f(x) for every x∈V(G)\{u,v}. We say that a w-dominating function f is a secure w-dominating function if for every v with f(v)=0, there exists u∈N(v) such that f(u)>0 and fu→v is a w-dominating function as well. The secure w-domi...

Bulletin of the Malaysian Mathematical Sciences Society, 2021

Let w = (w 0 , w 1 ,. .. , w l) be a vector of nonnegative integers such that w 0 ≥ 1. Let G be a... more Let w = (w 0 , w 1 ,. .. , w l) be a vector of nonnegative integers such that w 0 ≥ 1. Let G be a graph and N(v) the open neighbourhood of v ∈ V (G). We say that a function f : V (G) −→ {0, 1,. .. , l} is a w-dominating function if f (N(v)) = ∑ u∈N(v) f (u) ≥ w i for every vertex v with f (v) = i. The weight of f is defined to be ω(f) = ∑ v∈V (G) f (v). Given a w-dominating function f and any pair of adjacent vertices v, u ∈ V (G) with f (v) = 0 and f (u) > 0, the function f u→v is defined by f u→v (v) = 1, f u→v (u) = f (u) − 1 and f u→v (x) = f (x) for every x ∈ V (G) \ {u, v}. We say that a w-dominating function f is a secure w-dominating function if for every v with f (v) = 0, there exists u ∈ N(v) such that f (u) > 0 and f u→v is a w-dominating function as well. The (secure) w-domination number of G, denoted by (γ s w (G)) γ w (G), is defined as the minimum weight among all (secure) w-dominating functions. In this paper, we show how the secure (total) domination number and the (total) weak Roman domination number of lexicographic product graphs G • H are related to γ s w (G) or γ w (G). For the case of the secure domination number and the weak Roman domination number, the decision on whether w takes specific components will depend on the value of γ s (1,0) (H), while in the case of the total version of these parameters, the decision will depend on the value of γ s (1,1) (H).

Mathematics, Apr 15, 2020

In this article, we obtain general bounds and closed formulas for the secure total domination num... more In this article, we obtain general bounds and closed formulas for the secure total domination number of rooted product graphs. The results are expressed in terms of parameters of the factor graphs involved in the rooted product.

Applicable Analysis and Discrete Mathematics, 2018

In this paper we propose formulas for the distance between vertices of a generalized Sierpi?ski g... more In this paper we propose formulas for the distance between vertices of a generalized Sierpi?ski graph S(G, t) in terms of the distance between vertices of the base graph G. In particular, we deduce a recursive formula for the distance between an arbitrary vertex and an extreme vertex of S(G, t), and we obtain a recursive formula for the distance between two arbitrary vertices of S(G, t) when the base graph is triangle-free. From these recursive formulas, we provide algorithms to compute the distance between vertices of S(G, t). In addition, we give an explicit formula for the diameter and radius of S(G, t) when the base graph is a tree.

The Computer Journal, 2020

Let (X,d)(X,d)(X,d) be a metric space. A set SsubseteqXS\subseteq XSsubseteqX is said to be a kkk-metric generator for XXX ... more Let (X,d)(X,d)(X,d) be a metric space. A set SsubseteqXS\subseteq XSsubseteqX is said to be a kkk-metric generator for XXX if and only if for any pair of different points u,vinXu,v\in Xu,vinX, there exist at least kkk points w1,w2,ldotswkinSw_1,w_2, \ldots w_k\in Sw1,w2,ldotswkinS such that d(u,wi)ned(v,wi),;textrmforall;iin1,ldotsk.d(u,w_i)\ne d(v,w_i),\; \textrm{for all}\; i\in \{1, \ldots k\}.d(u,wi)ned(v,wi),;textrmforall;iin1,ldotsk. Let mathcalRk(X)\mathcal{R}_k(X)mathcalRk(X) be the set of metric generators for XXX. The kkk-metric dimension dimk(X)\dim _k(X)dimk(X) of (X,d)(X,d)(X,d) is defined as \begin{equation*}\dim_k(X)=\inf\{|S|:\, S\in \mathcal{R}_k(X)\}.\end{equation*}$$Here, we discuss the kkk-metric dimension of (V,dt)(V,d_t)(V,dt), where VVV is the set of vertices of a simple graph GGG and the metric dt:VtimesVrightarrowmathbbNcup0d_t:V\times V\rightarrow \mathbb{N}\cup \{0\}dt:VtimesVrightarrowmathbbNcup0 is defined by dt(x,y)=mind(x,y),td_t(x,y)=\min \{d(x,y),t\}dt(x,y)=mind(x,y),t from the geodesic distance ddd in GGG and a positive integer ttt. The case tgeD(G)t\ge D(G)tgeD(G), where D(G)D(G)D(G) denotes the diameter of GGG, corresponds to the original theory of kkk-metric dimension, and the case t=2t=2t=2 corresponds to the theory of kkk-adjacency dimension. Furthermore, t...

International Transactions in Operational Research, 2020

Theoretical Computer Science, 2020

As a generalization of the concept of the partition dimension of a graph, this article introduces... more As a generalization of the concept of the partition dimension of a graph, this article introduces the notion of the k-partition dimension. Given a nontrivial connected graph G = (V, E), a partition Π of V is said to be a k-partition generator of G if any pair of different vertices u, v ∈ V is distinguished by at least k vertex sets of Π, i.e., there exist at least k vertex sets S 1 ,. .. , S k ∈ Π such that d(u, S i) = d(v, S i) for every i ∈ {1,. .. , k}. A k-partition generator of G with minimum cardinality among all their k-partition generators is called a k-partition basis of G and its cardinality the k-partition dimension of G. A nontrivial connected graph G is k-partition dimensional if k is the largest integer such that G has a k-partition basis. We give a necessary and sufficient condition for a graph to be r-partition dimensional and we obtain several results on the k-partition dimension for k ∈ {1,. .. , r}.

Journal of the Operational Research Society, 2019

Discrete Applied Mathematics, 2019

The General Randić index R α of a simple graph G is defined as R α (G) = v i ∼v j (d(v i)d(v j)) ... more The General Randić index R α of a simple graph G is defined as R α (G) = v i ∼v j (d(v i)d(v j)) α , where d(v i) denotes the degree of the vertex v i. Rodríguez-Velázquez and Tomás-Andreu [MATCH Commun. Math. Comput. Chem. 74 (1) (2015) 145-160] obtained closed formulae for the Randić index R −1/2 of Sierpiński-type polymeric networks, where the base graph is a complete graph, a triangle-free regular graph or a bipartite semiregular graph. In the present article we obtain closed formulae for the general Randić index R α of Sierpiński-type polymeric networks, where the base graph is arbitrary.

ADVERTIMENT. L'accés als continguts d'aquesta tesi doctoral i la seva utilització ha de respectar... more ADVERTIMENT. L'accés als continguts d'aquesta tesi doctoral i la seva utilització ha de respectar els drets de la persona autora. Pot ser utilitzada per a consulta o estudi personal, així com en activitats o materials d'investigació i docència en els termes establerts a l'art. 32 del Text Refós de la Llei de Propietat Intel·lectual (RDL 1/1996). Per altres utilitzacions es requereix l'autorització prèvia i expressa de la persona autora. En qualsevol cas, en la utilització dels seus continguts caldrà indicar de forma clara el nom i cognoms de la persona autora i el títol de la tesi doctoral. No s'autoritza la seva reproducció o altres formes d'explotació efectuades amb finalitats de lucre ni la seva comunicació pública des d'un lloc aliè al servei TDX. Tampoc s'autoritza la presentació del seu contingut en una finestra o marc aliè a TDX (framing). Aquesta reserva de drets afecta tant als continguts de la tesi com als seus resums i índexs.