scholarly journals Some New Perspectives on Global Domination in Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-4 ◽  
Author(s):  
S. K. Vaidya ◽  
R. M. Pandit
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Affirmative Answer ◽  
Graph Operations ◽  
Domination In Graphs ◽  
Global Domination Number

A dominating set is called a global dominating set if it is a dominating set of a graph G and its complement G¯. Here we explore the possibility to relate the domination number of graph G and the global domination number of the larger graph obtained from G by means of various graph operations. In this paper we consider the following problem: Does the global domination number remain invariant under any graph operations? We present an affirmative answer to this problem and establish several results.

scholarly journals Some New Results on Global Dominating Sets

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
S. K. Vaidya ◽  
R. M. Pandit
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Affirmative Answer ◽  
Natural Question ◽  
Dominating Sets ◽  
Global Domination Number

A dominating set is called a global dominating set if it is a dominating set of a graph G and its complement G¯. A natural question arises: are there any graphs for which it is possible to relate the domination number and the global domination number? We have found an affirmative answer to this question and obtained some graphs having such characteristic.

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.  

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?

Complementary equitably totally disconnected equitable domination in graphs

2020 ◽  
pp. 2150043
Author(s):  
P. Nataraj ◽  
R. Sundareswaran ◽  
V. Swaminathan
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Domination In Graphs ◽  
Totally Disconnected

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.

Outer-paired domination in graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050072
Author(s):  
A. Mahmoodi ◽  
L. Asgharsharghi
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Paired Domination Number ◽  
Vertex Set ◽  
Paired Domination ◽  
Domination In Graphs

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. An outer-paired dominating set [Formula: see text] of a graph [Formula: see text] is a dominating set such that the subgraph induced by [Formula: see text] has a perfect matching. The outer-paired domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-paired dominating set of [Formula: see text]. In this paper, we study the outer-paired domination number of graphs and present some sharp bounds concerning the invariant. Also, we characterize all the trees with [Formula: see text].

EQUITABLE DOMINATION IN GRAPHS

2011 ◽  
Vol 03 (03) ◽  
pp. 311-321 ◽  
Author(s):  
A. ANITHA ◽  
S. ARUMUGAM ◽  
MUSTAPHA CHELLALI
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Domination In Graphs

Let D be a dominating set of a graph G = (V, E). For v ∈ D, let n1(v) = |N(v) ∩ (V - D)| and for w ∈ V - D, let n2(w) = |N(w) ∩ D|. Then D is called an equitable dominating set of type 1 if |n1(v1) - n1(v2)| ≤ 1 for all v1, v2 ∈ D and is called an equitable dominating set of type 2 if |n2(w1) - n2(w2)| ≤ 1 for all w1, w2 ∈ V - D. The minimum cardinality of an equitable dominating set of G of type 1 (type 2) is called the 1-equitable (2-equitable) domination number of G and is denoted by γ eq1 (G)(γ eq2 (G)). If D is an equitable dominating set of type 1 and type 2, then D is called an equitable dominating set and the equitable domination number of G is defined to be the minimum cardinality of an equitable dominating set and is denoted by γ eq (G). In this paper we initiate a study of these parameters.

scholarly journals On restricted domination in graphs

2007 ◽  
Vol 57 (5) ◽  
Author(s):  
Vladimir Samodivkin
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Minimum Number ◽  
Domination In Graphs ◽  
Restricted Domination Number

AbstractThe k-restricted domination number of a graph G is the minimum number d k such that for any subset U of k vertices of G, there is a dominating set in G including U and having at most d k vertices. Some new upper bounds in terms of order and degrees for this number are found.

scholarly journals On the upper geodetic global domination number of a graph

2020 ◽  
Vol 39 (6) ◽  
pp. 1627-1647
Author(s):  
X. Lenin Xaviour ◽  
S. Robinson Chellathurai
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
General Properties ◽  
Geodetic Set ◽  
Positive Integers ◽  
Global Domination Number

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.

Sign in / Sign up

Export Citation Format

Share Document