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.

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?

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].

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 Minimum Total Dominating Energy of Some Special Classes of Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 221-226
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Paley Graph ◽  
Eigen Values ◽  
Vertex Set

Let 𝑮 = (𝑽,𝑬) be a simple, finite, connected and undirected graph with vertex set V(G) and edge set E(G). Let 𝑺 ⊆ 𝑽(𝑮). A set S of vertices of G is a dominating set if every vertex in 𝑽 𝑮 − 𝑺 is adjacent to at least one vertex in S. A set S of vertices in a graph 𝑮(𝑽,𝑬) is called a total dominating set if every vertex 𝒗 ∈ 𝑽 is adjacent to an element of S. The minimum cardinality of a total dominating set of G is called the total domination number of G which is denoted by 𝜸𝒕 (𝑮). The energy of the graph is defined as the sum of the absolute values of the eigen values of the adjacency matrix. In this paper, we computed minimum total dominating energy of some special graphs such as Paley graph, Shrikhande graph, Clebsch graph, Chvatal graph, Moser graph and Octahedron graph.

Characterization of some classes of graphs with equal domination number and isolate domination number

2020 ◽  
Vol 12 (05) ◽  
pp. 2050065
Author(s):  
Davood Bakhshesh
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Interval Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Proper Interval Graph ◽  
Domination In Graphs

Let [Formula: see text] be a simple and undirected graph with vertex set [Formula: see text]. A set [Formula: see text] is called a dominating set of [Formula: see text], if every vertex in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text], denoted by [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is called isolate dominating, if the induced subgraph [Formula: see text] of [Formula: see text] contains at least one isolated vertex. The minimum cardinality of an isolate dominating set of [Formula: see text] is called the isolate domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we show that for every proper interval graph [Formula: see text], [Formula: see text]. Moreover, we provide a constructive characterization for trees with equal domination number and isolate domination number. These solve part of an open problem posed by Hamid and Balamurugan [Isolate domination in graphs, Arab J. Math. Sci. 22(2) (2016) 232–241].

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 Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{&#x0394;}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

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.

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.

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