Equitable Dominating Research Articles

Complementary Nil Equitable Domination in Fuzzy Graphs

Objectives: To demonstrate an inventive showcase prominent as complementary nil equitable dominating set, complementary nil equitable domination value in fuzzy graphs (FGs), and pertinent innovations. Methods: Strong arc in FGs ideas are used to perceive complementary nil equitable domination value. Findings: Modern concepts of complementary nil equitable dominance in FGs will lead to a number of applications, including the creation of cell phone towers in remote places, traffic light signals, and Google Maps. Novelty: Employing the ideas of equitable and strong arc in FGs, the complementary nil equitable dominating set and complementary nil equitable domination value in FGs are obtained. Some points are observed in this concept and some theorem related to the complementary nil equitable dominant set, its numbers are proved. AMS Subject Classification: 05C72. Keywords: Fuzzy graph, Equitable dominating set, Equitable domination number, Complementary nil equitable dominating set, Complementary nil equitable domination number

Outdegree equitable domination number of certain graph operators

For a vertex u in the dominating set D of a graph G, the number of edges from u to V-D is called the outdegree of u with respect to D, d°D, G(u). A dominating set D is called the outdegree equitable dominating set if the absolute value of the differences of outdegrees of any two vertices in D is at most one. The minimum cardinality of an outdegree equitable dominating set of G is called the outdegree equitable domination number of G, γoe(G). In this paper, we study the outdegree equitable domination number of certain graph operators such as complement, double graph, mycielskian and subdivision of graph.

On some independent equitable domination of graphs

One of the most interesting facts about the study of connectivity and working is in the area of domination. By this, we determine the independent equitable domination number of some graphs and some results. Moreover, we investigate graphs with the existence of independent equitable dominating set.

A linear time algorithm for minimum equitable dominating set in trees

Let G = (V, E) be a graph. A subset De of V is said to be an equitable dominating set if for every v ∈ V \ De there exists u ∈ De such that uv ∈ E and |deg(u) − deg(v)| ≤ 1, where, deg(u) and deg(v) denote the degree of the vertices u and v respectively. An equitable dominating set with minimum cardinality is called the minimum equitable dominating set and its cardinality is called the equitable domination number and it is denoted by γe. The problem of finding minimum equitable dominating set in general graphs is NP-complete. In this paper, we give a linear time algorithm to determine minimum equitable dominating set of a tree.

Equitable Domination in Vague Graphs With Application in Medical Sciences

Considering all physical, biological, and social systems, fuzzy graph (FG) models serve the elemental processes of all natural and artificial structures. As the indeterminate information is an essential real-life problem, which is mostly uncertain, modeling the problems based on FGs is highly demanding for an expert. Vague graphs (VGs) can manage the uncertainty relevant to the inconsistent and indeterminate information of all real-world problems, in which FGs possibly will not succeed in bringing about satisfactory results. In addition, VGs are a very useful tool to examine many issues such as networking, social systems, geometry, biology, clustering, medical science, and traffic plan. The previous definition restrictions in FGs have made us present new definitions in VGs. A wide range of applications has been attributed to the domination in graph theory for several fields such as facility location problems, school bus routing, modeling biological networks, and coding theory. Concepts from domination also exist in problems involving finding the set of representatives, in monitoring communication and electrical networks, and in land surveying (e.g., minimizing the number of places a surveyor must stand in order to take the height measurement for an entire region). Hence, in this article, we introduce different concepts of dominating, equitable dominating, total equitable dominating, weak (strong) equitable dominating, equitable independent, and perfect dominating sets in VGs and also investigate their properties by some examples. Finally, we present an application in medical sciences to show the importance of domination in VGs.

Complementary equitably totally disconnected equitable domination in graphs

In a simple, finite and undirected graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is said to be a degree equitable dominating set if for every [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the degree of [Formula: see text] in [Formula: see text]. The minimum cardinality of such a dominating set is denoted by [Formula: see text] and is called the equitable domination number of [Formula: see text]. In this paper, we introduce Complementary Equitably Totally Disconnected Equitable domination in graphs and obtain some interesting results. Also, we discuss some bounds of this new domination parameter.

Some More Results on Total Equitable Bondage Number of A Graph

The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γ(G-E0) > γ (G). An equitable dominating set D is called a total equitable dominating set if the induced subgraph < D > has no isolated vertices. The total equitable domination number γte(G) of G is the minimum cardinality of a total equitable dominating set of G. If γte(G) ≠ |V(G)| and <G-E0> contains no isolated vertices then the total equitable bondage number bte(G) of a graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γte(G-E0) > γte(G). In the present work we prove some characterizations and investigate total equitable bondage number of Ladder and degree splitting of path.

Neighborhood Connected Two -Outdegree Equitable Domination of the ϑ-Obrazom Graphs of P_n,G_n and H_n

Let G=(V,E) be a connected graph. A two-outdegree equitable dominating set D of a graph G is called the neighborhood connected two-outdegree equitable dominating set (nc2oe-set) if the induced sub graph <N(D)> is connected. The minimum cardinality of the minimal neighborhood connected two-outdegree equitable dominating set is called neighborhood connected two-outdegree equitable domination number of G and is denoted by . In this paper, we initiated the a study of this parameter and obtained for the -obrazom graphs of and . Also, we investigated the two-outdegree equitable domination number of the -obrazom graphs of and .

Total Equitable Bondage Number of a Graph

If for any total dominating set D with ν ∈ V (G) − D there exists a vertex u ∈ D such that uν ∈ E (G) and |d(ν)−d(u)| ≤ 1 then D is called the total equitable dominating set. The minimum cardinality of the total equitable dominating set is called the total equitable domination number denoted by γet (G). The bondage number b(G) of a nonempty graph G is the minimum cardinality among all sets of edges E0 ⊆ E(G) for which γ(G – E0) > γ(G). We introduced the concept of total equitable bondage number and proved several results.

Degree Equitable Regular Set Domination in Graphs

A set D of vertices in a graph $$G =(V,E)$$ is said to be regular set dominating set if for every set $$I \subseteq V-D$$ there exists a nonempty set $$S \subseteq D$$ such that $$\langle I \cup S \rangle $$ is regular. A subset $$D \subseteq V(G)$$ is called an equitable dominating set of a graph G if every vertex $$v \in V(G) {\setminus } D$$ has a neighbor $$u \in D$$ such that $$|d_G(u)-d_G(v)| \le 1$$ . An equitable dominating set D is a DERSD-dominating set of G if D is a regular set dominating set of G. The DERSD-domination number of G, denoted by $$\gamma _{rs}^e(G)$$ , is the minimum cardinality of a DERSD-dominating set of G. We initiate the study of DERSD-domination in graphs and obtain some sharp bounds. Finally, we show that the decision problem for determining $$\gamma _{rs}^e(G)$$ is NP-complete which gives a solution to the open problem posed by Sampathkumar and Pushpa Latha in their article Set domination in graphs, J Graph Theory 18(5):489–495, (1994).

On Total Domination in some Path Related Graphs

If G is a graph with vertex set V(G) then dominating set D subset V(G) is called total if every vertex of V(G) is adjacent to at least one vertex of D while it is called equitable if for every vertex u in D there exists a vertex v in V(G) - D such that the degree difference between these vertices is at most one. A dominating set which is both total and equitable is called total equitable dominating set. The minimum cardinality of total dominating set of G is called total domination number of G which is denoted by gamma_t(G). The total equitable domination number of G is the minimum cardinality of total equitable domination number of G which is denoted by gamma_t^e(G). We determine the exact values of total domination number as well as total equitable domination number of some path related graphs.

Triple connected two out degree equitable domination number

A subset D of V is said to be a triple connected two out degree equitable dominating set if D is two out degree equitable dominating set and the induced sub graph <D> is triple connected. The minimum cardinally taken over all triple connected two out degree equitable dominating set is called triple connected two out degree equitable domination number and denoted by γtc2oe (G). In this paper triple connected two out degree equitable domination number for some standard is determined

Equi independent equitable dominating sets in graphs

We introduce the concept of an equi independent equitable dominating set and define equi independent equitable domination number. We also investigate the graph families whose equi independent equitable domination number and equitable domination number are same.

On Equi Independent Equitable Dominating Sets in Graphs

The concept of equi independent equitable domination is the combination of two important concepts, namely independent domination and equitable domination. A subset $D$ of $V(G)$ is called an equitable dominating set if for every $v \in V(G)-D$, there exists a vertex $u \in D$ such that $uv \in E(G)$ and $|deg(u)-deg(v)|\leq 1$. A vertex subset $D$ is said to be equitable independent set if any two vertices of $D$ are either non adjacent or if adjacent then their degrees differ by atleast 2. An equitable dominating set $D$ is said to be an equi independent equitable dominating set if it is also equitable independent set. The equi independent equitable domination number $i^e$ is the minimum cardinality of an equi independent equitable dominating set.

On the Injective Equitable Domination of Graphs

A dominating set D in a graph G is called an injective equitable dominating set (Inj-equitable dominating set) if for every , there exists such that u is adjacent to v and . The minimum cardinality of such a dominating set is denoted by and is called the Inj-equitable domination number of G. In this paper, we introduce the injective equitable domination of a graph and study its relation with other domination parameters. The minimal injective equitable dominating set, the injective equitable independence number , and the injective equitable domatic number are defined.

Some Results on Equi Independent Equitable Dominating Sets in Graph

A subset D of V(G) is called an equitable dominating set if for every ? ? V(G) - D, there exists a vertex u ? D such that u? ? E(G) and | deg(u) – deg(?) | ? 1. A subset D of V(G) is called an equitable independent set if for any D of V(G), ? ? Ne(u) for all ? ? D - {u}where, Ne(u) = {? ? V(G)/? ? N(u), | deg(u) – deg(?) | ? 1}. An equitable dominating set D is said to be an equi independent equitable dominating set if it is also an equitable independent set. The minimum cardinality of an equi independent equitable dominating set is called equi independent equitable domination number which is denoted by ie. We investigated an equi independent equitable domination number for some special graphs.

Locating Equitable Domination and Independence Subdivision Numbers of Graphs

Let G = (V,E) be a simple, undirected, finite nontrivial graph. A non empty set DV of vertices in a graph G is a dominating set if every vertex in V-D is adjacent to some vertex in D. The domination number (G) is the minimum cardinality of a dominating set of G. A dominating set D is a locating equitable dominating set of G if for any two vertices u,wєV-D, N(u)∩D ≠ N(w)∩D, |N(u)∩D| ≠ |N(w)∩D|. The locating equitable domination number of G is the minimum cardinality of a locating equitable dominating set of G. The locating equitable domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the locating equitable domination number and is denoted by sdγle(G). The independence subdivision number sdβle(G) to equal the minimum number of edges that must be subdivided in order to increase the independence number. In this paper, we establish bounds on sdγle(G) and sdβle(G) for some families of graphs.

English

Let G = (V, E) be a graph. A set D  V (G) is equitable dominating set of G if  v  V – D  a vertex u  D such that uv  E (G) and |d(u) – d(v)|  1. A set D  V (G) is outer equitable dominating set if D is equitable dominating and is connected graph. The outer equitable connected domination number of G is the minimum cardinality of the outer-equitable connected dominating set of G and is denoted by oec(G). We introduce in this paper the concept of total outer equitable connected domination, exact values for some particular classes of graphs are found, some results on total outer equitable domination number are also established.

Independent transversal equitable domination in graphs

A set S ⊆ V of vertices in a graph G =( V, E) is called an equitable dominating set if for every vertex u in V − S there exists at least one vertex v in S adjacent to u and |deg(u) − deg(v) |≤ 1. An equitable dominating set which intersects every maximum independent set in G is called an independent transversal equitable dominating set. The minimum cardinality of an independent transversal equitable dominating set is called the independent transversal equitable domination number of G and is denoted by γite(G). In this paper we begin an investigation of this parameter.

Equitable Graph of a Graph

A subset D of V is called an equitable dominating set if for every v ∈ V — D there exists a vertex u ∈ D such that uv ∈ E(G) and |d(u) — d(v)| ≤ 1,where d(u) denotes the degree of vertex u and d(v) denotes the degree of vertex v. The minimum cardinality of such a dominating set is denoted by γe and is called the equitable domination number of G. Degree Equitable domination on Graphs were introduced in [10].This Paper aims at the study of a new concept called equitable graph ofa graph introduced by Prof. E. Sampathkumar. In this paper we show that there is relation between γ(Ge) and γe(G). Further results on the new parameter γ(Ge) are discussed.