scholarly journals Minimal and Upper Cost Effective Domination in Graphs

2021 ◽  
Vol 14 (2) ◽  
pp. 537-550
Author(s):  
Hearty Nuenay Maglanque ◽  
Ferdinand P. Jamil
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Cost Effective ◽  
Minimal Cost ◽  
Type Result ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
The Cost ◽  
Domination In Graphs

Given a connected graph $G$, we say that $S\subseteq V(G)$ is a cost effective dominating set in $G$ if, each vertex in $S$ is adjacent to at least as many vertices outside $S$ as inside $S$ and that every vertex outside $S$ is adjacent to at least one vertex in $S$. The minimum cardinality of a cost effective dominating set is the cost effective domination number of $G$. The maximum cardinality of a cost effective dominating set is the upper cost effective domination number of $G$, and is denoted by $\gamma_{ce}^+(G).$ A cost effective dominating set is said to be minimal if it does not contain a proper subset which is itself a cost effective dominating in $G$. The maximum cardinality of a minimal cost effective dominating set in a graph $G$ is the minimal cost effective domination number of $G$, and is denoted by $\gamma_{mce}(G)$. In this paper we provide bounds on upper cost effective domination number and minimal cost effective domination number of a connected graph G and characterized those graphs whose upper and minimal cost effective domination numbers are either $1, 2$ or $n-1.$ We also establish a Nordhaus-Gaddum type result for the introduced parameters and solve some realization problems.

scholarly journals Cost Effective Domination in the Join, Corona and Composition of Graphs

2019 ◽  
Vol 12 (3) ◽  
pp. 978-998
Author(s):  
Ferdinand P. Jamil ◽  
Hearty Nuenay Maglanque
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Cost Effective ◽  
Minimal Cost ◽  
Dominating Sets ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
The Cost ◽  
Domination Numbers

Let $G$ be a connected graph. A cost effective dominating set in a graph $G$ is any set $S$ of vertices in $G$ satisfying the condition that each vertex in $S$ is adjacent to at least as many vertices outside $S$ as inside $S$ and every vertex outside $S$ is adjacent to at least one vertex in $S$. The minimum cardinality of a cost effective dominating set is the cost effective domination number of $G$. The maximum cardinality of a cost effective dominating set is the upper cost effective domination number of $G$. A cost effective dominating set is said to be minimal if it does not contain a proper subset which is itself a cost effective dominating in $G$. The maximum cardinality of a minimal cost effective dominating set in a graph $G$ is the minimal cost effective domination number of $G$.In this paper, we characterized the cost effective dominating sets in the join, corona and composition of graphs. As direct consequences, we the bounds or the exact cost effective domination numbers, minimal cost effective domination numbers and upper cost effective domination numbers of these graphs were obtained.

scholarly journals Co-Secure Set Domination in Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 66-68
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Secure Set ◽  
Domination In Graphs

Throughout this paper, consider G = (V,E) as a connected graph. A subset D of V(G) is a set dominating set of G if for every M  V / D there exists a non-empty set N of D such that the induced sub graph <MUN> is connected. A subset D of the vertex set of a graph G is called a co-secure dominating set of a graph if D is a dominating set, and for each u' D there exists a vertex v'V / D such that u'v' is an edge and D \u'v' is a dominating set. A co-secure dominating set D is a co-secure set dominating set of G if D is also a set dominating set of G. The co-secure set domination number G s cs γ is the minimum cardinality of a co-secure set dominating set. In this paper we initiate the study of this new parameter & also determine the co-secure set domination number of some standard graphs and obtain its bounds.

scholarly journals Neighborhood connected perfect domination in graphs

2012 ◽  
Vol 43 (4) ◽  
pp. 557-562 ◽  
Author(s):  
Kulandai Vel M.P. ◽  
Selvaraju P. ◽  
Sivagnanam C.
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Domination In Graphs

Let $G = (V, E)$ be a connected graph. A set $S$ of vertices in $G$ is a perfect dominating set if every vertex $v$ in $V-S$ is adjacent to exactly one vertex in $S$. A perfect dominating set $S$ is said to be a neighborhood connected perfect dominating set (ncpd-set) if the induced subgraph $$ is connected. The minimum cardinality of a ncpd-set of $G$ is called the neighborhood connected perfect domination number of $G$ and is denoted by $\gamma_{ncp}(G)$. In this paper we initiate a study of this parameter.

Movable resolving domination in graphs

2021 ◽  
Author(s):  
Gerald B. Monsanto ◽  
Helen M. Rara
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Domination In Graphs

Let [Formula: see text] be a connected graph. Brigham et al., Resolving domination in graphs, Math. Bohem. 1 (2003) 25–36 defined a resolving dominating set as a set [Formula: see text] of vertices of a connected graph [Formula: see text] that is both resolving and dominating. A resolving dominating is a [Formula: see text]-movable resolving dominating set of [Formula: see text] if for every [Formula: see text], either [Formula: see text] is a resolving dominating set or there exists a vertex [Formula: see text] such that [Formula: see text] is a resolving dominating set of [Formula: see text]. The minimum cardinality of a [Formula: see text]-movable resolving dominating set of [Formula: see text], denoted by [Formula: see text] is the [Formula: see text]-movable[Formula: see text]-domination number of [Formula: see text]. A [Formula: see text]-movable resolving dominating set with cardinality [Formula: see text] is called a [Formula: see text]-set of [Formula: see text]. In this paper, we characterize the [Formula: see text]-movable resolving dominating sets in the join and lexicographic product of two graphs and determine the bounds or exact values of the [Formula: see text]-movable resolving domination number of these graphs.

scholarly journals Neighborhood connected edge domination in graphs

2012 ◽  
Vol 43 (1) ◽  
pp. 69-80
Author(s):  
Kulandaivel M.P. ◽  
C. Sivagnanam ◽  
P. Selvaraju
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Domination In Graphs

Let G = (V,E) be a connected graph. An edge dominating set X of G is called a neighborhood connected edge dominating set (nced-set) if the edge induced subgraph < N(X) > is connected. The minimum cardinality of a nced-set of G is called the neighborhood connected edge domination number of G and is denoted by. In this paper we initiate a study of this parameter.

scholarly journals Hop Domination in Graphs-II

2015 ◽  
Vol 23 (2) ◽  
pp. 187-199
Author(s):  
C. Natarajan ◽  
S.K. Ayyaswamy
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
The Family ◽  
Independent Domination ◽  
Domination In Graphs ◽  
Domination Numbers

Abstract Let G = (V;E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V - S, there exists u ∈ S such that d(u; v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γh(G). In this paper we characterize the family of trees and unicyclic graphs for which γh(G) = γt(G) and γh(G) = γc(G) where γt(G) and γc(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.

scholarly journals The Split Distance 2 Domination in Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 589
Author(s):  
A. Lakshmi ◽  
K. Ameenal Bibi ◽  
R. Jothilakshmi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Domination In Graphs

A distance - 2 dominating set D V of a graph G is a split distance - 2 dominating set if the induced sub graph <V-D> is disconnected. The split distance - 2 domination number is the minimum cardinality of a split distance - 2 dominating set. In this paper, we defined the notion of split distance - 2 domination in graph. We got many bounds on distance - 2 split domination number. Exact values of this new parameter are obtained for some standard graphs. Nordhaus - Gaddum type results are also obtained for this new parameter.  

scholarly journals Distance Domination and Distance Irredundance in Graphs

10.37236/953 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Adriana Hansberg ◽  
Dirk Meierling ◽  
Lutz Volkmann
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Distance Domination

A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]\neq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$. A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant. The $k$-irredundance number of $G$, denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemańska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.

Domination Theory in Graphs

2020 ◽  
pp. 1-23
Author(s):  
E. Sampathkumar ◽  
L. Pushpalatha
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Correct Conclusion ◽  
Minimum Number ◽  
Domination In Graphs

The study of domination in graphs originated around 1850 with the problems of placing minimum number of queens or other chess pieces on an n x n chess board so as to cover/dominate every square. The rules of chess specify that in one move a queen can advance any number of squares horizontally, vertically, or diagonally as long as there are no other chess pieces in its way. In 1850 enthusiasts who studied the problem came to the correct conclusion that all the squares in an 8 x 8 chessboard can be dominated by five queens and five is the minimum such number. With very few exceptions (Rooks, Bishops), these problems still remain unsolved today. Let G = (V,E) be a graph. A set S ⊂ V is a dominating set of G if every vertex in V–S is adjacent to some vertex in D. The domination number γ(G) of G is the minimum cardinality of a dominating set.

Pitchfork domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050025
Author(s):  
Manal N. Al-Harere ◽  
Mohammed A. Abdlhusein
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
New Model ◽  
Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

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