On restricted 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.

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Captive domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500767 ◽

2020 ◽

Vol 12 (06) ◽

pp. 2050076 ◽

Cited By ~ 1

Author(s):

Manal N. Al-Harere ◽

Ahmed A. Omran ◽

Athraa T. Breesam

Keyword(s):

Dominating Set ◽

Domination Number ◽

Upper Bounds ◽

Proper Subset ◽

Total Domination ◽

Lower And Upper Bounds ◽

Graph Domination ◽

Total Dominating Set ◽

Definition Of ◽

Domination In Graphs

In this paper, a new definition of graph domination called “Captive Domination” is introduced. The proper subset of the vertices of a graph [Formula: see text] is a captive dominating set if it is a total dominating set and each vertex in this set dominates at least one vertex which doesn’t belong to the dominating set. The inverse captive domination is also introduced. The lower and upper bounds for the number of edges of the graph are presented by using the captive domination number. Moreover, the lower and upper bounds for the captive domination number are found by using the number of vertices. The condition when the total domination and captive domination number are equal to two is discussed and obtained results. The captive domination in complement graphs is discussed. Finally, the captive dominating set of paths and cycles are determined.

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Weakly convex domination subdivision number of a graph

Filomat ◽

10.2298/fil1608101d ◽

2016 ◽

Vol 30 (8) ◽

pp. 2101-2110

Author(s):

Magda Dettlaff ◽

Saeed Kosary ◽

Magdalena Lemańska ◽

Seyed Sheikholeslami

Keyword(s):

Dominating Set ◽

Domination Number ◽

Upper Bounds ◽

Weakly Convex ◽

Minimum Cardinality ◽

Minimum Number ◽

Convex Domination Number ◽

Domination Subdivision Number

A set X is weakly convex in G if for any two vertices a,b ? X there exists an ab-geodesic such that all of its vertices belong to X. A set X ? V is a weakly convex dominating set if X is weakly convex and dominating. The weakly convex domination number ?wcon(G) of a graph G equals the minimum cardinality of a weakly convex dominating set in G. The weakly convex domination subdivision number sd?wcon (G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the weakly convex domination number. In this paper we initiate the study of weakly convex domination subdivision number and establish upper bounds for it.

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TOTAL OUTER-CONNECTED DOMINATION SUBDIVISION NUMBERS IN GRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830913500092 ◽

2013 ◽

Vol 05 (03) ◽

pp. 1350009

Author(s):

O. FAVARON ◽

R. KHOEILAR ◽

S. M. SHEIKHOLESLAMI

Keyword(s):

Connected Graph ◽

Dominating Set ◽

Domination Number ◽

Upper Bounds ◽

Connected Dominating Set ◽

Minimum Size ◽

Connected Domination Number ◽

Minimum Number ◽

Domination Subdivision Number

A set S of vertices of a graph G is a total outer-connected dominating set if every vertex in V(G) is adjacent to some vertex in S and the subgraph G[V\S] induced by V\S is connected. The total outer-connected domination numberγ toc (G) is the minimum size of such a set. The total outer-connected domination subdivision number sd γ toc (G) is the minimum number of edges that must be subdivided in order to increase the total outer-connected domination number. We prove the existence of sd γ toc (G) for every connected graph G of order at least 3 and give upper bounds on it in some classes of graphs.

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Hop Domination in Graphs-II

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.1515/auom-2015-0036 ◽

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.

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The Split Distance 2 Domination in Graphs

International Journal of Engineering & Technology ◽

10.14419/ijet.v7i4.10.21289 ◽

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.

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Pitchfork domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500251 ◽

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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Complementary equitably totally disconnected equitable domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500439 ◽

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.

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Outer-paired domination in graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s179383092050072x ◽

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].

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EQUITABLE DOMINATION IN GRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830911001231 ◽

2011 ◽

Vol 03 (03) ◽

pp. 311-321 ◽

Cited By ~ 2

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.

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