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The notion of Roman { 2 } -domination was introduced in 2016 as a variant of Roman domination, a concept inspired by a defending strategy used by the emperor Constantine (272–337 AD) to protect the Roman Empire. Since then, a considerable number of papers on Roman { 2 } -domination and its variants have been published. In this paper, we survey published results on Roman { 2 } -domination as well as the main findings on Roman { 2 } -domination variants found in the literature. A list of open problems related to this notion and its variants are also given. [ABSTRACT FROM AUTHOR]

Domination is an important factor in determining the robustness of a graph structure. A thorough examination of the graph's topological structure is necessary for analyzing and examining it for various aspects. Understanding the stability of a chemical compound is a significant criterion in chemistry, which necessitates conducting numerous experimental tests. The domination number and power domination number are pivotal in defining a wide range of physical properties, which include physiochemical properties, thermodynamic properties, chemical activities, and biological activities. The one-pentagonal carbon nanocone (1-PCNC) is a member of the carbon nanocone family and has a structure similar to that of honeycomb networks, which are renowned for their robustness. In this paper, we find the domination number and power domination number of 1-PCNC by considering it as an (m-1)-layered infinite graph. [ABSTRACT FROM AUTHOR]

Let G = (V (G), E(G)) be a simple and undirected graph. A dominating set S ⊆ V (G) is called a differentiating odd dominating set if for every vertex v ∈ V (G), |N[v] ∩ S| ≡ 1(mod 2) and NG[u] ∩ S≠ NG[v] ∩ S for every two distinct vertices u and v in V (G). The minimum cardinality of a differentiating odd dominating set of G, denoted by γo D(G), is called the differentiating odd domination number. In this paper, we discuss differentiating odd dominating sets and give bounds or exact values of the differentiating odd domination numbers of some graphs. We give necessary and sufficient conditions for some graphs to admit a differentiating odd dominating set. Moreover, we characterize the differentiating odd dominating sets in graphs resulting from join, corona, and lexicographic product of some graphs and determine the differentiating odd domination numbers of these graphs. [ABSTRACT FROM AUTHOR]

In this paper we study domination between different types of walks connecting two non-adjacent vertices of a graph. In particular, we center our attention on weakly toll walk and lk-path for k ∈ {2, 3}. A walk between two non-adjacent vertices in a graph G is called a weakly toll walk if the first and the last vertices in the walk are adjacent, respectively, only to the second and second-to-last vertices, which may occur more than once in the walk. And an lk-path is an induced path of length at most k between two nonadjacent vertices in a graph G. We study the domination between weakly toll walks, lk-paths (k ∈ {2, 3}) and different types of walks connecting two non-adjacent vertices u and v of a graph (shortest paths, induced paths, paths, tolled walks, weakly toll walks, lk-paths for k ∈ {3, 4}), and show how these give rise to characterizations of graph classes. [ABSTRACT FROM AUTHOR]

Consider a graph G with vertex set V(G) and edge set E(G). A subset D of V(G) is said to be a dominating set of G if every vertex not in D is adjacent to at least one vertex in D. If, in addition, every vertex not in a dominating set R of G is adjacent to at least one vertex in V (G) - R , then R is called a restrained dominating set of G. A paired restraint dominating set S of a graph G is a restrained dominating set of G satisfying that the induced subgraph by S contains a perfect matching. The problems of computing the dominating set, restrained dominating set, and paired restraint dominating set with minimum cardinality are referred to as the domination, restrained domination, and paired restraint domination problems, respectively. The paired restraint domination problem and its applications are first proposed here. Our focus is on examining the complexity of the proposed problem on extended supergrid graphs and their subclasses, which include grid, diagonal supergrid, and (original) supergrid graphs. The domination problem is known to be NP-complete on grid graphs and, therefore, also on extended supergrid graphs. Our previous research demonstrated that the domination and restrained domination problems on diagonal and original supergrid graphs are NP-complete. However, the complexity of the paired restraint domination problem on grid, diagonal supergrid, and original supergrid graphs remains unknown. The NP-completeness of the paired restraint domination problem on diagonal supergrid graphs is demonstrated in this paper, and this finding is also applicable to the original supergrid graphs and planar graphs with maximum degree 4. We then examine a subclass of diagonal and original supergrid graphs known as rectangular supergrid graphs. These graphs are distinguished by a rectangular shape consisting of m rows and n columns of vertices. Specifically, we address the paired restraint domination problem on R m × n and develop a linear time algorithm for 3 ⩾ m ⩾ 1 and n ⩾ m . Then, when n ⩾ m ⩾ 4 , we obtain a tight upper bound on the minimum size of the paired restraint dominating set of R m × n and then use this upper bound to establish one lower bound. [ABSTRACT FROM AUTHOR]

A graph G is a DTDP-graph if it has a pair (D, T) of disjoint sets of vertices of G such that D is a dominating set and T is a total dominating set of G. Such graphs were studied in a number of research papers. In this paper we study further properties of DTDP-graphs and, in particular, we characterize minimal DTDP-graphs without loops. [ABSTRACT FROM AUTHOR]

A set D of vertices in a graph G is a dominating set if every vertex not in D is adjacent to at least one vertex in D. The minimum cardinality of a dominating set in G is called the domination number and is denoted by γ(G). Let D be a minimum dominating set of G. If V-D contains a dominating set say D ′ of G, then D ′ is called an inverse dominating set with respect to D. The inverse domination number γ ′ (G) is the cardinality of a minimum inverse dominating set of G. A dominating set D is called a connected dominating set or an independent dominating set of G according as the induced subgraph ⟨D⟩ is connected or independent in G. The minimum of the cardinalities of the connected dominating sets of G or the independent dominating sets of G is called the connected domination number γc(G) or the independent domination number γi(G) respectively. In this paper, we determine the inverse domination numbers in X-Trees and Sibling Trees. We have also determined the independent domination numbers of both the trees and the connected domination number of Sibling Trees. A result on inverse domination number of some classes of Hypertrees is also included. [ABSTRACT FROM AUTHOR]

A set C of vertices in a graph G = (V, E) is an identifying code if it is dominating and any two vertices of V are dominated by distinct sets of codewords. This paper presents a survey of Iiro Honkala's contributions to the study of identifying codes with respect to several aspects: complexity of computing an identifying code, combinatorics in binary Hamming spaces, infinite grids, relationships between identifying codes and usual parameters in graphs, structural properties of graphs admitting identifying codes, and number of optimal identifying codes. [ABSTRACT FROM AUTHOR]

In this paper, we consider a variant of dominating set problem, i.e., the total dominating set problem. Given an undirected graph G = (V , E) , a subset of vertices T ⊆ V is called a total dominating set if every vertex in V is adjacent to at least one vertex in T. Based on LP relaxation techniques, this paper gives a distributed approximation algorithm for the total dominating set problem in general graphs. The presented algorithm obtains a fractional total dominating set that is, at most, k (1 + Δ 1 k) Δ 1 k times the size of the optimal solution to this problem, where k is a positive integer and Δ is the maximum degree of G. The running time of this algorithm is constant communication rounds under the assumption of a synchronous communication model. [ABSTRACT FROM AUTHOR]

In an existential-phenomenological approach, the world is not narrowed down to external things and events but is understood as a context of meaning that includes everything that people experience and 'in which' they conduct their lives. This paper undertakes an analysis of how digital technology is transforming the world in an existential-phenomenological sense and why such a digitally transformed world diminishes the human capacity for religious experience. In accord with Husserl's characterisation of science as a mathematical project of nature and with Heidegger's critique of technology as the will to power, digital technology represents a further step in subjugating the world and life. Such a mastering stance, however, not only closes man off from the possibility of experiencing the transcendent 'otherness' that forms the basis for religious experience, but also closes him off from authentic life, which is inconceivable without elements of uncontrollability and contingency. Overcoming our digital entrapment requires a recognition of the role of the 'lifeworld', which, through its passive elements, enables us to experience gratitude and makes us receptive to religious experience. [ABSTRACT FROM AUTHOR]

A subset D of the vertex set V (G) in a graph G is a point-set dominating set (or, in short, psd-set) of G if for every set S ⊆ V - D, there exists a vertex v ∈ D such that the induced subgraph hS [fvgi is connected. The minimum cardinality of a psd-set of G is called the point-set domination number of G. In this paper, we establish two sharp lower bounds for point-set domination number of a graph in terms of its diameter and girth. We characterize graphs for which lower bound of point set domination number is attained in terms of its diameter. We also establish an upper bound and give some classes of graphs which attains the upper bound of point set domination number. [ABSTRACT FROM AUTHOR]

For a graph G with chromatic number k, a dominating set S of G is called a chromatic-transversal dominating set (ctd-set) if S intersects every color class of any k-coloring of G. The minimum cardinality of a ctd-set of G is called the chromatic transversal domination number of G and is denoted by γct(G). A Roman dominating function (RDF) in a graph G is a function f: V (G) → {0, 1, 2} satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value w(f) =Σu∈V f(u). The minimum weight of a Roman dominating function of a graph G is called the Roman domination number of G and is denoted by γR(G). The concept of chromatic transversal domination is extended to Roman domination as follows: For a graph G with chromatic number k, a Roman dominating function f is called a chromatic-transversal Roman dominating function (CTRDF) if the set of all vertices v with f(v) > 0 intersects every color class of any k-coloring of G. The minimum weight of a chromatic-transversal Roman dominating function of a graph G is called the chromatic-transversal Roman domination number of G and is denoted by γctR(G). In this paper a study of this parameter is initiated. [ABSTRACT FROM AUTHOR]

Domination plays an indispensable role in graph theory. Various types of domination explore various types of applications. Equal-status people work together and interlace with each other easily. In this paper, the paired equitable domination of a graph, its inflated graph, and its complement of an inflated graph were studied. The relationship between the domination number of the graph, the equitable domination number, and the paired equitable domination number of complements of the inflated graph were established. Furthermore, we proved the Nordhaus–Gaddum-type inequality, that is, γ p r e (H) + γ p r e (H) ≤ 6 if H is a graph with m nodes where m ≡ 0 ,   2 (m o d   8) and d (a i) = (m/2) for all a i . The challenges and limitations of this parameter of paired equitable and equitable domination depends on the degree of the vertex of the graph. Practical applications are discussed in various fields and illustrated using the studied parameter. [ABSTRACT FROM AUTHOR]

In this paper, we have initiated the study of domination and equitable domination of Neutrosophic graphs using strong arcs. Strong arcs represent the optimal (minimum) degree of truth membership value, the optimal (minimum) degree of indeterminacy membership value, and the non-optimal ( maximum) degree of falsity membership value. Hence, the studies of domination and equitable domination using strong arcs have been explored. Upper bounds and minimality conditions for the existence of the introduced parameters were discussed. Extend the studies on strong and weak equitable domination of Neutrosophic graphs and obtain the relationship between domination and the equitable domination parameter. Furthermore, we have provided some theorems based on equitable domination of Neutrosophic graphs and discussed the upper and lower bounds of the strong and weak equitable domination in terms of order and size with other existing domination parameters of Neutrosophic graphs. [ABSTRACT FROM AUTHOR]

This paper analyzes a contemporary conception of digital despotism through themes drawn from classical Greek philosophy. By taking as a measure some of the most radically excluded categories of human existence, Aristotle’s slave and slavish types, I offer a way to understand digital despotism as a syndrome of overlapping risks to human impairment, brought about by the advent of automated data processing technologies, which dispossesses people along i) ontological and ii) cognitive dimensions. This conception aims to balance the appeal to the language of slavery in recent global historical, Marxist, republican, and postcolonial discourses on digital technology, while distinguishing itself from the coercive, material violence involved in the experiences of slavery itself. Unlike prior conceptions, this thematic idea of digital despotism moreover suggests political vulnerability to forms of despotic rule and integrates various risk factors that can therefore be better recognized in both policy intervention, and individual and/or collective resistance. [ABSTRACT FROM AUTHOR]

Let G = (V;E) be a graph of order n and size m: The graph Sp(G) obtained from G by adding a new vertex v0 for every vertex v 2 V and joining v0 to all neighbors of v in G is called the splitting graph of G: In this paper, we determine the domination number, the total domination number, connected domination number, paired domination number and independent domination number for the splitting graph Sp(G):. [ABSTRACT FROM AUTHOR]

In a simple, finite undirected graph G, a dominating set D is a subset of the vertex set V(G) whose closed neighbourhood is V(G). Many types of domination have been studied. The studies are based either on the nature of domination or the type of dominating set or the type of the complement of the dominating set. Interaction between dominating set and its complement is also considered. Fair domination is the domination where every vertex in the complement of a dominating set has equal number of neighbours in the dominating set. In this paper, a dominating set whose vertices have equal number of neighbours in the complement is the subject of study. The parameter γcof(G) is introduced and studied. [ABSTRACT FROM AUTHOR]

There has been an increased focus within the AI ethics literature on questions of power, reflected in the ideal of accountability supported by many Responsible AI guidelines. While this recent debate points towards the power asymmetry between those who shape AI systems and those affected by them, the literature lacks normative grounding and misses conceptual clarity on how these power dynamics take shape. In this paper, I develop a workable conceptualization of said power dynamics according to Cristiano Castelfranchi's conceptual framework of power and argue that end-users depend on a system's developers and users, because end-users rely on these systems to satisfy their goals, constituting a power asymmetry between developers, users and end-users. I ground my analysis in the neo-republican moral wrong of domination, drawing attention to legitimacy concerns of the power-dependence relation following from the current lack of accountability mechanisms. I illustrate my claims on the basis of a risk-prediction machine learning system, and propose institutional (external auditing) and project-specific solutions (increase contestability through design-for-values approaches) to mitigate domination. [ABSTRACT FROM AUTHOR]

In a graph G, a vertex is said to 2-step dominate itself and all the vertices which are at distance 2 from it in G. A set D of vertices in G is called a hop dominating set of G if every vertex outside D is 2-step dominated by some vertex of D. Given a graph G and a positive integer k, the hop domination problem is to decide whether G has a hop dominating set of cardinality at most k. The hop domination problem is known to be NP-complete for bipartite graphs. In this paper, we design a linear time algorithm for computing a minimum hop dominating set in chordal bipartite graphs. [ABSTRACT FROM AUTHOR]

The notion of fuzzy graph (FG) is widely used in many problems arising from partial or incomplete descriptions of the real world and in particular from fields such as engineering, economics, computer science, social disciplines, or medical diagnostics, and has been used in many fields of pure mathematics as well as in several areas of applied sciences such as decision making, statistics and networking. In this paper we will deal with the graph of the picture fuzzy(symmetric) set using the notion of domination in picture fuzzy graph (PFG) as a generalization of both the concept of fuzzy graph domination and intuitionistic fuzzy graph (IFG) domination. The concepts of domination theory (DT) and double domination theory (DDT) of a PFG are introduced, studied and concretely applied to the real case of an election competition to determine the minimum number of citizens a politician should meet in person in order to win the election. The choice of fuzzification (symmetric) and defuzzification (anti-symmetric) methods depends on the specific application and the type of fuzzy sets being used, whether they are symmetric or anti-symmetric. There are various methods for each process, such as centroid, max-min, and weighted average methods for defuzzification. Finally, in the last section, drawing from the application example, the features and benefits of PFGs with respect to fuzzy graphs and intuitionistic fuzzy graphs are compared and discussed. [ABSTRACT FROM AUTHOR]

Let G be a graph and S ⊆ V (G) represent a subset of vertices having installed "detectors," each of which is capable of sensing an "intruder" in its open-neighborhood. The open-locating-code of v ∈2 V (G) is the set of neighboring detectors, N(v) ∩ S. The set S is said to be an open-locating- dominating set if every open-locating-code is unique and non-empty. In this paper we focus on error-detecting open-locating-dominating sets on the in_nite triangular grid, present a solution with density 1/2, and prove it is optimal. [ABSTRACT FROM AUTHOR]

In 2013 the concept of [ j , k ] -dominating sets in graphs have been introduced as an extension of [ 1 , k ] -dominating sets. Regarding studying [ j , k ] -dominating sets, the research is also interesting if the additional independence requirement is assumed, because the existence of independent [ 1 , k ] -dominating sets is an NP -complete problem in general case. Inspired by this topic, in this paper we study the problem of the existence of independent [ j , k ] -dominating sets in a special, multi-argument graph product. We give the complete characterization of the generalized corona G ∘ h n with an independent [ j , k ] -dominating set, for arbitrary positive integers j and k, which generalize the known result for a corona of two graphs. [ABSTRACT FROM AUTHOR]

A coalition in a graph G = (V,E) consists of two disjoint sets V1 and V2 of vertices, such that neither V1 nor V2 is a dominating set, but the union V1 ∪ V2 is a dominating set of G. A coalition partition in a graph G of order n = |V | is a vertex partition π = {V1, V2, . . ., Vk} such that every set Vi either is a dominating set consisting of a single vertex of degree n - 1, or is not a dominating set but forms a coalition with another set Vj which is not a dominating set. Associated with every coalition partition π of a graph G is a graph called the coalition graph of G with respect to π, denoted CG(G, π), the vertices of which correspond one-to-one with the sets V1, V2, . . ., Vk of π and two vertices are adjacent in CG(G, π) if and only if their corresponding sets in π form a coalition. The singleton partition π1 of the vertex set of G is a partition of order |V |, that is, each vertex of G is in a singleton set of the partition. A graph G is called a self-coalition graph if G is isomorphic to its coalition graph CG(G, π1), where π1 is the singleton partition of G. In this paper, we characterize self-coalition graphs. [ABSTRACT FROM AUTHOR]

This paper argues that regulatory agencies have a responsibility to further the public interest when they determine the conditions under which new technological products may be commercialized. As a case study, this paper analyzes the US 9th Circuit Court's ruling on the efforts of the US Environmental Protection Agency (EPA) to regulate an herbicide meant for use with seed that are genetically modified to be tolerant of the chemical. Using that case, it is argued that when regulatory agencies evaluate new technological products, they have an obligation to draw on data, analyses, and evaluations from a variety of credible epistemic sources, and not rely solely or even primarily on the technology developer. Otherwise, they create conditions for their own domination and that of the polity by the technology developer. Moreover, in the interest of advancing the public interest, regulatory agencies must evaluate new technologies in a substantively and procedurally unbiased manner. [ABSTRACT FROM AUTHOR]

A set S ⊆ V (G) is a hop dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2. It is a global hop dominating set of G if it is a hop dominating set of both G and the complement G of G. The minimum cardinality of a hop dominating (global hop dominating) set of G, denoted by γh(G) (resp. γgh(G)), is called the hop domination (resp. global hop domination) number of G. In this paper, we give some realization results involving domination, hop domination, and global hop domination parameters. Also, we give a rectification of a result found in a recent paper of the authors and use this to prove some results in this paper. [ABSTRACT FROM AUTHOR]

Since 2016, when the first paper of the double Roman domination appeared, the topic has received considerable attention in the literature. We survey known results on double Roman domination and some variations of the double Roman domination, and a list of open questions and conjectures is provided. [ABSTRACT FROM AUTHOR]

A set S ⊆ V (G) of an undirected graph G is a locating-dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S ̸= NG(y) ∩ S for any two distinct vertices x and y in V (G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γSLD(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for edge corona and lexicographic product of graphs. [ABSTRACT FROM AUTHOR]

For a given graph G=(V,E), a Roman {2}-dominating function f : V (G) → {0, 1, 2} has the property that for every vertex u with f(u) = 0, either u is adjacent to a vertex assigned 2 under f, or is adjacent to at least two vertices assigned 1 under f. The Roman {2}-domination number of G, {R2}(G), is the minimum of P u∈V (G) f(u) over all such functions. In this paper, we initiate the study of the problem of finding Roman {2}-bondage number of G. The Roman {2}-bondage number of G, b{R2}, is defined as the cardinality of a smallest edge set E′ ⊆ E for which {R2}(G-E′) > {R2}(G). We first demonstrate complexity status of the problem by proving that the problem is NP-Hard. Then, we derive useful parametric as well as fixed upper bounds on the Roman {2}-bondage number of G. Specifically, it is known that the Roman bondage number of every planar graph does not exceed 15 (see [S. Akbari, M. Khatirinejad and S. Qajar, A note on the Roman bondage number of planar graphs, Graphs Combin. 29 (2013) 327-331]). We show that same bound will be preserved while computing the Roman {2}-bondage number of such graphs. The paper is then concluded by computing exact value of the parameter for some classes of graphs. [ABSTRACT FROM AUTHOR]

For a graph G = (V,E), a Roman {2}-dominating function (R2DF) f: V = {0, 1, 2} has the property that for every vertex v = V with f(v) = 0, either there exists a neighbor u = N(v), with f(u) = 2, or at least two neighbors x, y = N(v) having f(x) = f(y) = 1. The weight of an R2DF f is the sum f(V) = P v2V f(v), and the minimum weight of an R2DF on G is the Roman {2}-domination number {R2}(G). An R2DF is independent if the set of vertices having positive function values is an independent set. The independent Roman {2}-domination number i{R2}(G) is the minimum weight of an independent Roman {2}-dominating function on G. In this paper, we show that the decision problem associated with {R2}(G) is NP- complete even when restricted to split graphs. We design a linear time algorithm for computing the value of i{R2}(T) in any tree T, which answers an open problem raised by Rahmouni and Chellali [Independent Roman {2}- domination in graphs, Discrete Appl. Math. 236 (2018) 408-414]. Moreover, we present a linear time algorithm for computing the value of {R2}(G) in any block graph G, which is a generalization of trees. [ABSTRACT FROM AUTHOR]

Let G be an undirected graph with vertex and edge sets V (G) and E(G), respectively. A set S ⊆ V (G) is a hop independent set of G if any two distinct vertices in S are not at a distance two from each other, that is, dG(v, w) ≠ 2 for any distinct vertices v, w ∈ S. The maximum cardinality of a hop independent set of G, denoted by αh(G), is called the hop independence number of G. In this paper, we show that the absolute difference of the independence number and the hop independence number of a graph can be made arbitrarily large. Furthermore, we determine the hop independence numbers of some graphs including those resulting from some binary operations of graphs. [ABSTRACT FROM AUTHOR]

A set D of vertices in a graph G is a dominating set of G if every vertex outside D is adjacent in G to some vertex in D. A set D of vertices in G is a semitotal dominating set of G if D is a dominating set of G and every vertex in D is within distance 2 from another vertex of D. Given a graph G and a positive integer k, the semitotal domination problem is to decide whether G has a semitotal dominating set of cardinality at most k. The semitotal domination problem is known to be NP-complete for chordal graphs and bipartite graphs as shown in [M.A. Henning and A. Pandey, Algorithmic aspects of semitotal domination in graphs, Theoret. Comput. Sci. 766 (2019) 46-57]. In this paper, we present a linear time algorithm to compute a minimum semitotal dominating set in block graphs. On the other hand, we show that the semitotal domination problem remains NP-complete for undirected path graphs. [ABSTRACT FROM AUTHOR]

In this paper, we provide a structural characterization of graphs having a spanning tree with disjoint dominating and 2-dominating sets. [ABSTRACT FROM AUTHOR]

Strictly speaking, Bourdieu did not develop specific works on emotions and affects. However, as we expect to show, the emotional dimension is an integral part of his perspective and a gateway to his reflections on power. With this premise, the objective of this paper is to analyze Bourdieu's perspective on symbolic power, based on the notion of illusio. Can illusio be understood as a mechanism of power and symbolic domination involving forms of emotional expression? Based on the analysis carried out, we support an affirmative answer to this question, assuming illusio as a principle of perception, emotional investment, adhesion and belief, by means of which the agents affectively commit themselves to the asymmetric logic of the fields in which they participate, moved by aspirational mechanisms from which they end up indirectly legitimizing the social order. [ABSTRACT FROM AUTHOR]

Given a graph G and an integer k, it is an NP-complete problem to decide whether G has a dominating set of size at most k. In this paper we study this problem for the Knödel Graph on n vertices using elementary number theory techniques. In particular, we show an explicit upper bound for the domination number of the Knödel Graph on n vertices any time that we can find a prime number p dividing n for which 2 is a primitive root. [ABSTRACT FROM AUTHOR]

This article explores interrelations between artists, environment and rural gentrification, through researches conducted in England and France. In these geographical contexts, it will be argued that artists emphasize socio-spatial changes and emerging socio-environmental inequalities. Firstly, the paper highlights multiple links between artists and rural and protected areas, in particular spatial forms of artistic establishment and their effects in terms of gentrification. Secondly, the paper seeks to deconstruct the role of artists in rural gentrification. Either pioneers or catalysts of the process, artists can locally boost social and spatial forms of domination. For some artists, gentrification can embody an opportunity from a professional point of view. Therefore, through their creations, artists can exacerbate tensions and support new forms of exclusion. Nevertheless, art proves to be a relevant tool to freshen dialogue within gentrified countryside. [ABSTRACT FROM AUTHOR]

Abstract Let G = (V , E) be a graph and t , r be positive integers. The reception strength (or signal) that a vertex v receives from a tower of signal strength t located at vertex T is defined as s i g (v , T) = m a x (t − d i s t (v , T) , 0) , where d i s t (v , T) denotes the distance between the vertices v and T. In 2015 Blessing, Insko, Johnson, and Mauretour defined a (t , r) broadcast dominating set , or simply a (t , r) broadcast , on G as a set T ⊆ V such that the sum of all signals received at each vertex v ∈ V is at least r. We say that T is optimal if | T | is minimal among all such sets T. The cardinality of an optimal (t , r) broadcast on a finite graph G is called the (t , r) broadcast domination number of G. The concept of (t , r) broadcast domination generalizes the classical problem of domination on graphs. In fact, the (2 , 1) broadcasts on a graph G are exactly the dominating sets of G. In their paper, Blessing et al. considered (t , r) ∈ { (2 , 2) , (3 , 1) , (3 , 2) , (3 , 3) } and gave optimal (t , r) broadcasts on G m , n , the grid graph of dimension m × n , for small values of m and n. They also provided upper bounds on the optimal (t , r) broadcast numbers for grid graphs of arbitrary dimensions. In this paper, we define the density of a (t , r) broadcast, which allows us to provide optimal (t , r) broadcasts on the infinite grid graph for all t ≥ 2 and r = 1 , 2 , and bound the density of the optimal (t , 3) broadcast for all t ≥ 2. In addition, we present a Python program to compute upper bounds on the density of a minimal (t , r) broadcast on the infinite grid, and compute these bounds for all 1 ≤ t ≤ 15 and 1 ≤ r ≤ 40. Lastly, we construct a family of counterexamples to the conjecture of Blessing et al. that the optimal (t , r) and (t + 1 , r + 2) broadcasts are identical for all t ≥ 1 and r ≥ 1 on the infinite grid. [ABSTRACT FROM AUTHOR]

Crimediscusses this author's analyses of crime and urban sociability. Reviewing Machado da Silva's papers and book chapters on the subject, the text highlights how both the methodological turn towards everyday life and the theoretical reframing of the domination question support the innovative sociological interpretation the sociologist provides about urban violence in Brazil. The paper stresses that this analytical approach follows Machado da Silva's previous research on urban lower classes' daily lives. [ABSTRACT FROM AUTHOR]

A subset D ⊆ VG is a dominating set of G if every vertex in VG -- D has a neighbor in D, while D is a paired-dominating set of G if D is a dominating set and the subgraph induced by D contains a perfect matching. A graph G is a DPDP-graph if it has a pair (D; P) of disjoint sets of vertices of G such that D is a dominating set and P is a paired-dominating set of G. The study of the DPDP-graphs was initiated by Southey and Henning [Cent. Eur. J. Math. 8 (2010) 459-467; J. Comb. Optim. 22 (2011) 217-234]. In this paper, we provide conditions which ensure that a graph is a DPDP-graph. In particular, we characterize the minimal DPDP-graphs. [ABSTRACT FROM AUTHOR]

A set S ⊆ V (G) of a (simple) undirected graph G is a locating-dominating set of G if for each v∈V (G) n S, there exists w∈S such tha vw∈E(G) and NG(x) \ S 6= NG(y) \ S for any distinct vertices x and y in V (G) n S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S n fvg is a locating-dominating set of G for each v∈S. The minimum cardinality of a stable locating-dominating set of G, denoted by s l (G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for some graphs. Further, we introduce other related concepts and use them to characterize the stable locating-dominating sets in some graphs. [ABSTRACT FROM AUTHOR]

A total Roman dominating function on a graph G is a function f: V (G) ! f0; 1; 2g such that for every vertex v 2 V (G) with f(v) = 0 there exists a vertex u 2 V (G) adjacent to v with f(u) = 2, and the subgraph induced by the set fx 2 V (G): f(x) = 1g has no isolated vertices. The total Roman domination number of G, denoted tR(G), is the minimum weight !(f) = P v2V (G) f(v) among all total Roman dominating functions f on G. It is known that tR(G) = t2(G) + (G) for any graph G with neither isolated vertex nor components isomorphic to K2, where t2(G) and (G) represent the semitotal domination number and the classical domination number, respectively. In this paper we give a constructive characterization of the trees that satisfy the equality above. [ABSTRACT FROM AUTHOR]

A self-complementary graph is a graph isomorphic to its complement. A set S of vertices in a graph G is a restrained dominating set if every vertex in V (G) \ S is adjacent to a vertex in S and to a vertex in V (G) \ S. The restrained domination number of a graph G is the minimum cardinality of a restrained dominating set of G. In this paper, we study restrained domination in self-complementary graphs. In particular, we characterize the self-complementary graphs having equal domination and restrained domination numbers. [ABSTRACT FROM AUTHOR]

The vulnerability in a communication network is the measurement of the strength of the network against damage that occurs in nodes or communication links. It is important that a communication network is still effective even when it loses some of its nodes or links. In other words, since a network can be modelled by a graph, it is desired to know whether the graph is still connected when some of the vertices or edges are removed from a connected graph. The vulnerability parameters aim to find the nature of the network when a subset of the nodes or links is removed. One of these parameters is domination. Domination is a measure of the connection of a subset of vertices with its complement. In this paper, we study porous exponential domination as a vulnerability parameter and obtain certain results on the Cartesian product and lexicographic product graphs. We determine the porous exponential domination number, denoted by , of the Cartesian product of with and , separately. We also determine the porous exponential domination number of the Cartesian product of with complete bipartite graphs and any graph which has a vertex of degree . Moreover, we obtain the porous exponential domination number of the lexicographic product of and , denoted by , for the case where is a graph of order with a vertex of degree and for the opposite case where is a graph of order which has no vertex of degree . We further show that by proving , where is a graph of order with a vertex of degree and is any graph of order . [ABSTRACT FROM AUTHOR]

In the context of the recent struggle for the right to abortion in Argentina, the slogan of “separation of Church and State” emerged strongly within a sector of the women’s movement. This elementary liberal demand, formulated in the twenty-first century, stimulates reflection on the scope of secularization in modern societies and on the relations between the political and religious spheres. As a result, this article recovers the theoretical legacy of Max Weber, who not only made the universal process of rationalization his privileged research problem, but also carefully approached the categories of State and Church and the link that these institutions stablished historically. In that sense, the paper aims to: 1) examine the way in which Weber constructs his definitions of these two sociological concepts, specifying their affinities; 2) identify the most important typicalideal relationships that he sees between the two institutions in the pre-modern world; and 3) characterize the separation of Church and State in modernity and the limitations that he finds in this area. [ABSTRACT FROM AUTHOR]

A set S ⊆ V is a dominating set in G if for every u ∈ V \ S, there exists v ∈ S such that (u, v) ∈ E, i.e., N[S] = V. A dominating set S is an isolate dominating set (IDS) if the induced subgraph G[S] has at least one isolated vertex. It is known that Isolate Domination Decision problem (IDOM) is NP-complete for bipartite graphs. In this paper, we extend this by showing that the IDOM is NP-complete for split graphs and perfect elimination bipartite graphs, a subclass of bipartite graphs. A set S ⊆ V is an independent set if G[S] has no edge. A set S ⊆ V is a secure dominating set of G if, for each vertex u ∈ V \ S, there exists a vertex v ∈ S such that (u, v) ∈ E and (S \ {v})∪ {u} is a dominating set of G. In addition, we initiate the study of a new domination parameter called, independent secure domination. A set S ⊆ V is an independent secure dominating set (InSDS) if S is an independent set and a secure dominating set of G. The minimum size of an InSDS in G is called the independent secure domination number of G and is denoted by γis(G). Given a graph G and a positive integer k, the InSDM problem is to check whether G has an independent secure dominating set of size at most k. We prove that InSDM is NP-complete for bipartite graphs and linear time solvable for bounded tree-width graphs and threshold graphs, a subclass of split graphs. The MInSDS problem is to find an independent secure dominating set of minimum size, in the input graph. Finally, we show that the MInSDS problem is APX-hard for graphs with maximum degree 5. [ABSTRACT FROM AUTHOR]

Let G be a graph and γ (G) denote the domination number of G , i.e. the cardinality of a smallest set of vertices S such that every vertex of G is either in S or adjacent to a vertex in S. Matheson and Tarjan conjectured that a plane triangulation with a sufficiently large number n of vertices has γ (G) ≤ n ∕ 4. Their conjecture remains unsettled. In the present paper, we show that: (1) a maximal outerplane graph with n vertices has γ (G) ≤ ⌈ n + k 4 ⌉ where k is the number of pairs of consecutive degree 2 vertices separated by distance at least 3 on the boundary of G ; and (2) a Hamiltonian plane triangulation G with n ≥ 23 vertices has γ (G) ≤ 5 n ∕ 16. We also point out and provide counterexamples for several recent published results of Li et al. (2016) on this topic which are incorrect. [ABSTRACT FROM AUTHOR]

The problem of monitoring an electric power system by placing as few measurement devices in the system can be formulated as a power dominating set problem in graph theory. The power domination number of a graph is the minimum cardinality of a power dominating set. Xu and Kang [On the power domination number of the generalized Petersen graphs, J. Comb. Optim. 22 (2011) 282-291] study the exact power domination number for the generalized Petersen graph P(3k, k), and propose the following problem: determine the power domination number for the generalized Petersen graph P(4k, k) or P(ck, k). In this paper we give the power domination number for P(4k, k) and present a sharp upper bound on the power domination number for the generalized Petersen graph P(ck, k). [ABSTRACT FROM AUTHOR]

For a graph G=(V,E), a set S⊆V is a [1,2]-set if it is a dominating set for G and each vertex v∈V∖S is dominated by at most two vertices of S, i.e. 1≤|N(v)∩S|≤2. Moreover a set S⊆V is a total [1,2]-set if for each vertex of V, it is the case that 1≤|N(v)∩S|≤2. The [1,2]-domination number of G, denoted γ[1,2](G),is the minimum number of vertices in a [1,2]-set. Every [1,2]-set with cardinality of γ[1,2](G) is called a γ[1,2]-set. Total [1,2]-domination number and γt[1,2]-sets of G are defined in a similar way. This paper presents a linear time algorithm to find a γ[1,2]-set and a γt[1,2]-set in generalized series-parallel graphs. [ABSTRACT FROM AUTHOR]

A set D of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to a vertex in D and the subgraph induced by D contains a perfect matching (not necessarily as an induced subgraph). A paired-dominating set of G is minimal if no proper subset of it is a paired-dominating set of G. The upper paired-domination number of G , denoted by Γ pr (G) , is the maximum cardinality of a minimal paired-dominating set of G. In Upper-PDS , it is required to compute a minimal paired-dominating set with cardinality Γ pr (G) for a given graph G. In this paper, we show that Upper-PDS cannot be approximated within a factor of n 1 − ε for any ε > 0 , unless P = NP and Upper-PDS is APX -complete for bipartite graphs of maximum degree 4. On the positive side, we show that Upper-PDS can be approximated within O (Δ) -factor for graphs with maximum degree Δ. We also show that Upper-PDS is solvable in polynomial time for threshold graphs, chain graphs, and proper interval graphs. [ABSTRACT FROM AUTHOR]

A perfect Roman dominating function on a graph G is a function f : V (G) ⟶ { 0 , 1 , 2 } having the property that for every vertex u with f (u) = 0 , there exists exactly one vertex v such that u v ∈ E (G) and f (v) = 2. The weight of f , denoted by w (f) , is the value ∑ v ∈ V (G) f (v). Given a graph G and a positive integer k , the perfect Roman domination problem is to decide whether there is a perfect Roman dominating function f on G such that w (f) is at most k. In this paper, we first show that the perfect Roman domination problem is NP -complete for chordal graphs, planar graphs, and bipartite graphs. Then we present polynomial time algorithms for computing a perfect Roman dominating function with minimum weight in block graphs, cographs, series-parallel graphs, and proper interval graphs. • We study the perfect Roman domination problem from the algorithmic point of view. • The perfect Roman domination problem is NP -complete for chordal graphs, planar graphs, and bipartite graphs. • We present linear time algorithms for computing the perfect Roman domination number in block graphs, cographs, and series-parallel graphs. • Further, a polynomial time algorithm is proposed for computing the perfect Roman domination number in proper interval graphs. [ABSTRACT FROM AUTHOR]

In this paper, we characterize cubic and quartic Cayley graphs on abelian groups that admit an efficient dominating set, and then we give domination numbers, even when they do not have an efficient dominating set, when it is possible. [ABSTRACT FROM AUTHOR]