Total Near Equitable Domination in Graphs (original) (raw)
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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.
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
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.
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.
Connected Equitable Domination in Graphs
m-hikari.com
Let G = (V, E) be a graph. A subset D of V is called an equitable dominating set of a graph G if for every v ∈ V − D, there exists a vertex v ∈ D such that uv ∈ E(G) and |deg (u)
Degree equitable restrained double domination in graphs
2021
A subset D ⊆ V ( G ) is called an equitable dominating set of a graph G if every vertex v ∈ V ( G ) \ D has a neighbor u ∈ D such that | d G ( u )- d G ( v )| ≤ 1. An equitable dominating set D is a degree equitable restrained double dominating set (DERD-dominating set) of G if every vertex of G is dominated by at least two vertices of D , and 〈 V ( G ) \ D 〉 has no isolated vertices. The DERD-domination number of G , denoted by γ cl ^ e ( G ), is the minimum cardinality of a DERD-dominating set of G . We initiate the study of DERD-domination in graphs and we obtain some sharp bounds. Finally, we show that the decision problem for determining γ cl ^ e ( G ) is NP-complete.
arXiv (Cornell University), 2011
A fair dominating set in a graph G (or FD-set) is a dominating set S such that all vertices not in S are dominated by the same number of vertices from S; that is, every two vertices not in S have the same number of neighbors in S. The fair domination number, fd(G), of G is the minimum cardinality of a FD-set. We present various results on the fair domination number of a graph. In particular, we show that if G is a connected graph of order n ≥ 3 with no isolated vertex, then fd(G) ≤ n − 2, and we construct an infinite family of connected graphs achieving equality in this bound. We show that if G is a maximal outerplanar graph, then fd(G) < 17n/19. If T is a tree of order n ≥ 2, then we prove that fd(T) ≤ n/2 with equality if and only if T is the corona of a tree.
arXiv (Cornell University), 2017
A set S ⊆ V is a dominating set of G if every vertex in V − S is adjacent to at least one vertex in S. The domination number γ(G) of G equals the minimum cardinality of a dominating set S in G; we say that such a set S is a γ-set. The single greatest focus of research in domination theory is the determination of the value of γ(G). By definition, all vertices must be dominated by a γ-set. In this paper we propose relaxing this requirement, by seeking sets of vertices that dominate a prescribed fraction of the vertices of a graph. We focus particular attention on 1/2 domination, that is, sets of vertices that dominate at least half of the vertices of a graph G.