Connected Equitable Domination in Graphs (original) (raw)

Connected Near Equitable Domination in Graphs

2013

Let G = (V;E) be a graph, D ⊆ V and u be any vertex in D. Then the out degree of u with respect to D denoted by odD(u), is dened as odD(u) = |N(u) ∩ (V − D)|. A subset D ⊆ V (G) is called a near equitable dominating set of G if for every v ∈ V − D there exists a vertex u ∈ D such that u is adjacent to v and |odD(u) − odV D(v)| 6 1. A near equitable dominating set D is said to be a connected near equitable dominating set if the subgraph ⟨D⟩ induced by D is connected. The minimum of the cardinality of a connected near equitable dominating set of G is called the connected near equitable domination number and is denoted by cne(G). In this paper results involving this parameter are found, bounds for cne(G) are obtained. Connected near equitable domatic partition in a graph G is studied.

The Split equitable domination Number of a Graph

An equitable dominating set D of a graph G = (V, E) is a split equitable dominating set if the induced subgraph hV − Di is a disconnected. The split equitable domination number ϒse(G) of a graph G is the minimum cardinality of a split equitable dominating set. In this paper, we initiate the study of this new parameter and present some bounds and some exact values for ϒse(G). Also Nordhaus−Gaddum type results are obtained.

Equitable Edge Domination in Graphs

2012

A subset D of V (G) is called an equitable dominating set of a graph G if for every v ∈ (V − D), there exists a vertex u ∈ D such that uv ∈ E(G) and |deg(u) − deg(v)| 6 1. The minimum cardinality of such a dominating set is denoted by γe(G) and is called equitable domination number of G. In this paper we introduce the equitable edge domination and equitable edge domatic number in a graph, exact value for the some standard graphs bounds and some interesting results are obtained.

Neighborhood Connected Equitable Domination in Graphs

Applied Mathematical Sciences, 2012

Let G = (V, E) be a connected graph, An equitable dominating S of a graph G is called the neighborhood connected equitable dominating set (nced-set) if the induced subgraph N e (S) is connected The minimum cardinality of a nced-set of G is called the neighborhood connected equitable domination number of G and is denoted by γ nce

Total Near Equitable Domination in Graphs

2014

Let G = (V,E) be a graph, DV and u be any vertex in D. Then the out degree of u with respect to D denoted by odD(u), is defined as odD(u) = |N(u) \ (V D)|. A subset DV (G) is called a near equitable dominating set of G if for every v 2 V D there exists a vertex u 2 D such that u is adjacent to v and |odD(u) odV −D(v)| � 1. A near equitable dominating set D is said to be a total near equitable dominating set (tned-set) if every vertex w 2 V is adjacent to an element of D. The minimum cardinality of tned-set of G is called the total near equitable domination number of G and is denoted by tne(G). The maximum order of a partition of V into tned-sets is called the total near equitable domatic number of G and is denoted by dtne(G). In this paper we initiate a study of these

Two-Out Degree Equitable Domination in Graphs

An equitable domination has interesting application in the context of social networks. In a network, nodes with nearly equal capacity may interact with each other in a better way. In the society persons with nearly equal status, tend to be friendly. In this paper, we introduce new variant of equitable domination of a graph. Basic properties and some interesting results have been obtained. G is called the domination number (upper domination number) of G and is denoted by γ(G) (Γ(G)). An excellent treatment of the fundamentals of domination is given in the book by Haynes et al. [4]. A survey of several advanced topics in domination is given in the book edited by Haynes et al. [5]. Various types of domination have been defined and studied by several authors and more than 75 models of domination are listed in the appendix of Haynes et al. [4]. A double star is the tree obtained from two disjoint stars K 1,n and K 1,m by connecting their centers.

Triple connected two out degree equitable domination number

Journal of Discrete Mathematical Sciences and Cryptography

In this paper we introduce a new domination parameter with real life application called neighborhood triple connected two out degree equitable domination number of a graph. A subset D of V of a nontrivial graph G is said to be a neighborhood triple connected two out degree equitable dominating set if D is a two out degree equitable dominating set and the induced sub graph <N(D)> is triple connected. The minimum cardinality taken over all neighborhood triple connected two out degree equitable dominating set is called neighborhood triple connected two out degree equitable domination number and is denoted by 2 () We investigate this number for some standard graphs and special graphs

On Connected Partial Domination in Graphs

European Journal of Pure and Applied Mathematics, 2021

This paper introduces and investigates a variant of partial domination called the connected α-partial domination. For any graph G = (V (G), E(G)) and α ∈ (0, 1], a set S ⊆ V (G) is an α-partial dominating set in G if |N[S]| ≥ α |V (G)|. An α-partial dominating set S ⊆ V (G) is a connected α-partial dominating set in G if ⟨S⟩, the subgraph induced by S, is connected. The connected α-partial domination number of G, denoted by ∂Cα(G), is the smallest cardinality of a connected α-partial dominating set in G. In this paper, we characterize the connected α-partial dominating sets in the join and lexicographic product of graphs for any α ∈ (0, 1] and determine the corresponding connected α-partial domination numbers of graphs resulting from the said binary operations. Moreover, we establish sharp bounds for the connected α-partial domination numbers of the corona and Cartesian product of graphs. Furthermore, we determine ∂Cα(G) of some special graphs when α...

Super Equitable Domination in Graphs

Advances in Mathematics: Scientific Journal

An equitable dominating set D of V (G) is called a super equitable dominating set of G if every vertex of V − D has a private equitable neighbour in D. This paper initiates the study of super equitable dominating set.