Global Dominating Research Articles

Algorithms for the global domination problem

A dominating set D in a graph G is a subset of its vertices such that every its vertex that does not belong to set D is adjacent to at least one vertex from set D. A set of vertices of graph G is a global dominating set if it is a dominating set for both, graph G and its complement. The objective is to find a global dominating set with the minimum cardinality. Neither exact nor approximation algorithm existed for the problem known to be NP-hard. We show that it remains NP-hard for restricted types of graphs. At the same time, we specify some families of graphs for which the three heuristics, that we propose here, are optimal. Given the complexity status of the problem, our aim was the development of powerful heuristic algorithms that work well in practice for large-scaled instances. To measure the efficiency of our heuristics, we formulated the problem as an integer linear program (ILP) and also we developed an alternative implicit enumeration (IE) algorithm obtaining guaranteed optimal solutions for the existing benchmark instances with up to 8000 vertices. Remarkably, for 56.75% of these instances, at least one of our heuristics also created an optimal solution, where an average absolute error for the remaining instances was a single vertex. The average approximation ratio was 1.005, whereas for the largest benchmark instances with up to 25000 vertices our heuristics delivered solutions in less than 2 min.

The forcing geodetic global domination number of a graph

Let [Formula: see text] be a connected graph and [Formula: see text] be a minimum geodetic global dominating set of [Formula: see text]. A subset [Formula: see text] is called a forcing subset for [Formula: see text] if [Formula: see text] is the unique minimum geodetic global dominating set containing [Formula: see text]. A forcing subset for [Formula: see text] of minimum cardinality is a minimum forcing subset of [Formula: see text]. The forcing geodetic global domination number of [Formula: see text], denoted by [Formula: see text], is the cardinality of a minimum forcing subset of S. The forcing geodetic global domination number of [Formula: see text], denoted by [Formula: see text], is [Formula: see text], where the minimum is taken over all minimum geodetic global dominating sets [Formula: see text] in [Formula: see text]. The forcing geodetic global domination number of some standard graphs are determined. Some of its general properties are studied. It is shown that for every pair of positive integers a and b with [Formula: see text] and [Formula: see text], there exists a connected graph [Formula: see text] such that [Formula: see text] and [Formula: see text]. The geodetic global domination number of join of graphs is also studied. Connected graphs of order [Formula: see text] with geodetic global domination number 2 are characterized. It is proved that, for a connected graph [Formula: see text] with [Formula: see text]. Then [Formula: see text] and characterized connected graphs for which the lower and the upper bounds are sharp.

Russia has Spread its Wings. Phenomenology of our Era Global Systematic Contradiction Development

The article reveals specific features of the present day stage in the global systematic contradiction ‘creditors – debtors’ development. The origin of its critical stage was depicted in the context of form variety in the total hybrid, informational war that was projected well in advance, thoroughly worked-out and unleashed in a mass way against Russia by overseas and over-oceans Anglo-Saxon political elites. Technologies of the present day informational wars were studied and tools of cynical manipulation of public opinion of people in their own countries used to obtain a mass support of any decisions made by their governments were described. The massive information flow is meant to form a smoke-screen, which could hide real goals and intentions of architects of global domination. Deliberately formulated lies aim at discrediting a previously chosen object in eyes of the global public. This object is a nominated institutional enemy. This object forms a barrier due to its natural being and it is absolutely unacceptable for a new idea, i.e. global dominating exalted by British political elite to the strategic imperative of this implicit, pressing demand. Strategic imperative has a direct meaning, it is an official doctrine presented in a solemn, high-society atmosphere by the head of the Ministry of Foreign Affairs of Great Britain in the London City. The implicit demand, forming the essence of strategic imperative arose, grew and was promoted by all possible ways. This demand makes today the essence of Anglo-Saxon elites’ interests, the essence of any form of overthrowing the enemy in the process of developing the global systematic social-economic antagonism ‘creditors-debtors’. Today contradiction of interests of social groups forming poles of this antagonism passed over to the stage of global systematic antagonism of our era and has reached its climax. To attain the longed-for goal of strategic imperative for all of us London is going to stick to a new approach in international relations, which will be based on three spheres: military force, economic security and more advanced global alliances. Strategy of British political elite was defined and declared in public. An open, unprecedented by its aims and attracted resources challenge was thrown down to institutional enemies. Russia has accepted the challenge. Today Russia is facing numerous most serious problems that must be settled without any doubts.

Detour global domination for splitting graph

In this paper, we introduced the new concept detour global domination number for splitting graph of standard graph. The detour global dominating sets in some standard and special graphs are determined. First we recollect the concept of splitting graph of a graph and we produce some results based on the detour global domination number of splitting graph of star graph, bistar graph and complete bipartite graph. A set [Formula: see text] is called a detour global dominating set of [Formula: see text] if [Formula: see text] is both detour and global dominating set of [Formula: see text]. The detour global domination number is the minimum cardinality of a detour global dominating set in [Formula: see text].

Global dominating broadcast in graphs

A broadcast on a graph [Formula: see text] is a function [Formula: see text] such that for each [Formula: see text], where [Formula: see text] denotes the diameter of [Formula: see text] and [Formula: see text] denotes the eccentricity of [Formula: see text]. The cost of a broadcast is the value [Formula: see text]. In this paper, we define and study a new invariant of broadcasts in graphs, which is global dominating broadcast. A dominating broadcast [Formula: see text] of a graph [Formula: see text] is a global dominating broadcast if [Formula: see text] is also a dominating broadcast of [Formula: see text] the complement of [Formula: see text]. We begin by determining the global broadcast domination number of bipartite graphs, paths, cycles, grid graphs and trees. Then we establish lower and upper bounds on the global broadcast domination number of a graph. Finally, we establish relationships between the global broadcast domination number and other parameters.

Revisiting Domination, Hop Domination, and Global Hop Domination in Graphs

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 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. Â

Connected Geodetic Global Domination Number of a Graph

A set S of vertices in a connected graph {G=(V,E)} is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbour in D. A geodetic dominating set S is both a geodetic and a dominating set. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. The geodetic global domination number (geodetic domination number) is the minimum cardinality of a geodetic global dominating set (geodetic dominating set) in G. In this paper we introduced and investigate the connected geodetic global domination number of certain graphs and some of the general properties are studied.

The Connected Geodetic Global Domination Number of a Graph

A set S of vertices in a connected graph {G=(V,E)} is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbour in D. A geodetic dominating set S is both a geodetic and a dominating set. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. The geodetic global domination number (geodetic domination number) is the minimum cardinality of a geodetic global dominating set (geodetic dominating set) in G. In this paper we introduced and investigate the connected geodetic global domination number of certain graphs and some of the general properties are studied.

A Survey on Domination in Vague Graphs with Application in Transferring Cancer Patients between Countries

Many problems of practical interest can be modeled and solved by using fuzzy graph (FG) algorithms. In general, fuzzy graph theory has a wide range of application in various fields. Since indeterminate information is an essential real-life problem and is often uncertain, modeling these problems based on FG is highly demanding for an expert. A vague graph (VG) can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems in which fuzzy graphs may not succeed in bringing about satisfactory results. Domination in FGs theory is one of the most widely used concepts in various sciences, including psychology, computer sciences, nervous systems, artificial intelligence, decision-making theory, etc. Many research studies today are trying to find other applications for domination in their field of interest. Hence, in this paper, we introduce different kinds of domination sets, such as the edge dominating set (EDS), the total edge dominating set (TEDS), the global dominating set (GDS), and the restrained dominating set (RDS), in product vague graphs (PVGs) and try to represent the properties of each by giving some examples. The relation between independent edge sets (IESs) and edge covering sets (ECSs) are established. Moreover, we derive the necessary and sufficient conditions for an edge dominating set to be minimal and show when a dominance set can be a global dominance set. Finally, we try to explain the relationship between a restrained dominating set and a restrained independent set with an example. Today, we see that there are still diseases that can only be treated in certain countries because they require a long treatment period with special medical devices. One of these diseases is leukemia, which severely affects the immune system and the body’s defenses, making it impossible for the patient to continue living a normal life. Therefore, in this paper, using a dominating set, we try to categorize countries that are in a more favorable position in terms of medical facilities, so that we can transfer the patients to a suitable hospital in the countries better suited in terms of both cost and distance.

The monophonic global domination number of a graph

A set M ⊆ V is said to be a monophonic global dominating set of G if M is both a monophonic set and a global dominating set of G. The minimum cardinality of a monophonic global dominating set of G is the monophonic global domination number of G and is denoted by γm(G). A monophonic global dominating set of cardinality γm(G) is called a γm-set of G. The monophonic global domination number of certain classes of graphs are determined. It is proved that 2 ≤ γm(G) ≤ γg (G) ≤ n, where γg (G) is a geodetic global domination number of a G. It is shown that for every pair of positive integers a and b with 2 ≤ a ≤ b, there exists a connected graph G such that γm(G) = a and γg (G) = b.

On the upper geodetic global domination number of a graph

A set S of vertices in a connected graph G = (V, E) is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbor in D. A set D is called a global dominating set in G if S is a dominating set of both G and Ḡ. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. A geodetic global dominating set S in G is called a minimal geodetic global dominating set if no proper subset of S is itself a geodetic global dominating set in G. The maximum cardinality of a minimal geodetic global dominating set in G is the upper geodetic global domination number Ῡg+(G) of G. In this paper, the upper geodetic global domination number of certain connected graphs are determined and some of the general properties are studied. It is proved that for all positive integers a, b, p where 3 ≤ a ≤ b < p, there exists a connected graph G such that Ῡg(G) = a, Ῡg+(G) = b and |V (G)| = p.

Geodetic global domination in corona and strong product of graphs

A set S of vertices in a connected graph [Formula: see text] is called a geodetic set if every vertex not in [Formula: see text] lies on a shortest path between two vertices from [Formula: see text]. A set [Formula: see text] of vertices in [Formula: see text] is called a dominating set of [Formula: see text] if every vertex not in [Formula: see text] has at least one neighbor in [Formula: see text]. A set [Formula: see text] is called a geodetic global dominating set of [Formula: see text] if [Formula: see text] is both geodetic and global dominating set of [Formula: see text]. The geodetic global domination number is the minimum cardinality of a geodetic global dominating set in [Formula: see text]. In this paper, we determine the geodetic global domination number of the corona and strong products of two graphs.

Geodetic Global Domination in the Join of Two Graphs

A set S of vertices in a connected graph is called a geodetic set if every vertex not in lies on a shortest path between two vertices from . A set of vertices in is called a dominating set of if every vertex not in has at least one neighbor in . A set is called a geodetic global dominating set of if is both geodetic and global dominating set of . The geodetic global dominating number is the minimum cardinality of a geodetic global dominating set in . In this paper we determine the geodetic global domination number of the join of two graphs.

Outer independent global dominating set of trees and unicyclic graphs

Let G be a graph. A set D ⊆ V ( G ) is a global dominating set of G if D is a dominating set of G and $\overline G$ . γ g ( G ) denotes global domination number of G . A set D ⊆ V ( G ) is an outer independent global dominating set (OIGDS) of G if D is a global dominating set of G and V ( G ) − D is an independent set of G . The cardinality of the smallest OIGDS of G , denoted by γ g o i ( G ) , is called the outer independent global domination number of G . An outer independent global dominating set of cardinality γ g o i ( G ) is called a γ g o i -set of G . In this paper we characterize trees T for which γ g o i ( T ) = γ ( T ) and trees T for which γ g o i ( T ) = γ g ( T ) and trees T for which γ g o i ( T ) = γ o i ( T ) and the unicyclic graphs G for which γ g o i ( G ) = γ ( G ) , and the unicyclic graphs G for which γ g o i ( G ) = γ g ( G ) .

A generalization of global dominating function

Let $G$ be a graph‎. ‎A function $f‎ : ‎V (G) longrightarrow {0,1}$‎, ‎satisfying‎ ‎the condition that every vertex $u$ with $f(u) = 0$ is adjacent with at‎ ‎least one vertex $v$ such that $f(v) = 1$‎, ‎is called a dominating function $(DF)$‎. ‎The weight of $f$ is defined as $wet(f)=Sigma_{v in V(G)} f(v)$‎. ‎The minimum weight of a dominating function of $G$‎ ‎is denoted by‎ ‎$gamma (G)$‎, ‎and is called the domination number of $G$‎. ‎A dominating‎ ‎function $f$ is called a global dominating function $(GDF)$ if $f$ is‎ ‎also a $DF$ of $overline{G}$‎. ‎The minimum weight of a global dominating function is denoted by‎ ‎$gamma_{g}(G)$ and is called global domination number of $G$‎. ‎In this paper we introduce a generalization of global dominating function‎. ‎Suppose $G$ is a graph and $sgeq 2$ and $K_n$ is the complete graph on $V(G)$‎. ‎A function $ f:V(G)longrightarrow { 0,1} $ on $G$ is $s$-dominating function $(s-DF)$‎, ‎if there exists some factorization ${G_1,ldots,G_s }$ of $K_n$‎, ‎such that $G_1=G$ and $f$ is dominating function of each $G_i$‎.

Global Domination in Bipolar Fuzzy Graphs

In this paper the concept of global domination in bipolar fuzzy graph is introduced and studied its characterization. The necessary and sufficient condition for the global dominating set is given in terms of domination between the vertices. Also semi complete, purely semi complete, semi complementary and semi global domination concepts are defined and some results are obtained.

Global Domination Integrity of Graphs

R. Sundareswaran and V. Swaminathan [11], introduced the concept of the domination integrity of a graph. It is a useful measure of vulnerability and it is defined as DI(G) = min{|S|+m(G−S), where S is a dominating set and m(G−S) is the order of a maximum component of G−S}. In this paper we introduce the concept of global domination integrity of a graph G, and define it as GDI(G) = min{|S|+m(G−S), where S is a global dominating set and m(G−S) is the order of a maximum component of G−S}. The global domination integrity of some graphs is obtained. The relations between global domination integrity and other parameters are determined.

Global Roman domination in graphs

A Roman dominating function (RDF) on a graph G=(V,E) is defined to be a function f:V→{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. A set S⊆V is a global dominating set if S dominates both G and its complement G¯. The global domination number γg(G) of a graph G is the minimum cardinality of S. We define a global Roman dominating function on a graph G=(V,E) to be a function f:V→{0,1,2} such that f is an RDF for both G and its complement G¯. The weight of a global Roman dominating function is the value f(V)=∑u∈Vf(u). The minimum weight of a global Roman dominating function on a graph G is called the global Roman domination number of G and denoted by γgR(G). In this paper, we initiate a study of this parameter.

Semi Global Dominating Set in Fuzzy Graphs

Semi Global Dominating Set in Fuzzy Graphs

Bounds on the Global Domination Number

A set S of vertices in a graph G is a global dominating set of G if S simultaneously dominates both G and its complement Ḡ. The minimum cardinality of a global dominating set of G is the global domination number of G. We determine bounds on the global domination number of a graph and relationships between it and other domination related parameters.