Critical concepts of restrained domination in signed graphs

2021 ◽  
pp. 2250010
Author(s):  
Anisha Jean Mathias ◽  
V. Sangeetha ◽  
Mukti Acharya
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Signed Graph ◽  
Signed Graphs ◽  
Minimum Cardinality ◽  
Underlying Graph ◽  
Restrained Domination ◽  
Edge Removal ◽  
Restrained Domination Number

A signed graph [Formula: see text] is a simple undirected graph in which each edge is either positive or negative. Restrained dominating set [Formula: see text] in [Formula: see text] is a restrained dominating set of the underlying graph [Formula: see text] where the subgraph induced by the edges across [Formula: see text] and within [Formula: see text] is balanced. The minimum cardinality of a restrained dominating set of [Formula: see text] is called the restrained domination number, denoted by [Formula: see text]. In this paper, we initiate the study on various critical concepts to investigate the effect of edge removal or edge addition on restrained domination number in signed graphs.

scholarly journals Restrained domination in signed graphs

2020 ◽  
Vol 12 (1) ◽  
pp. 155-163
Author(s):  
Anisha Jean Mathias ◽  
V. Sangeetha ◽  
Mukti Acharya
Keyword(s):  
Social Networks ◽  
Dominating Set ◽  
Domination Number ◽  
Important Application ◽  
Signed Graph ◽  
Signed Graphs ◽  
Underlying Graph ◽  
Restrained Domination ◽  
Restrained Domination Number

AbstractA signed graph Σ is a graph with positive or negative signs attatched to each of its edges. A signed graph Σ is balanced if each of its cycles has an even number of negative edges. Restrained dominating set D in Σ is a restrained dominating set of its underlying graph where the subgraph induced by the edges across Σ[D : V \ D] and within V \ D is balanced. The set D having least cardinality is called minimum restrained dominating set and its cardinality is the restrained domination number of Σ denoted by γr(Σ). The ability to communicate rapidly within the network is an important application of domination in social networks. The main aim of this paper is to initiate a study on restrained domination in the realm of different classes of signed graphs.

Domination in signed graphs

2020 ◽  
pp. 2050094
Author(s):  
P. Jeyalakshmi
Keyword(s):  
Social Science ◽  
Dominating Set ◽  
Domination Number ◽  
Signed Graph ◽  
Signed Graphs ◽  
Minimum Cardinality ◽  
Underlying Graph

Let [Formula: see text] be a graph. A signed graph is an ordered pair [Formula: see text] where [Formula: see text] is a graph called the underlying graph of [Formula: see text] and [Formula: see text] is a function called a signature or signing function. Motivated by the innovative paper of B. D. Acharya on domination in signed graphs, we consider another way of defining the concept of domination in signed graphs which looks more natural and has applications in social science. A subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if [Formula: see text] for all [Formula: see text]. The domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a dominating set of [Formula: see text]. Also, a dominating set [Formula: see text] of [Formula: see text] with [Formula: see text] is called a [Formula: see text]-set of [Formula: see text]. In this paper, we initiate a study on this 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?

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.

scholarly journals Restrained Edge Domination Number of Some Path Related Graphs

2021 ◽  
Vol 13 (1) ◽  
pp. 145-151
Author(s):  
S. K. Vaidya ◽  
P. D. Ajani
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Restrained Domination ◽  
Edge Dominating Set ◽  
Middle Graph

For a graph G = (V,E), a set  S ⊆ V(S ⊆ E) is a restrained dominating (restrained edge dominating) set if every vertex (edge) not in S is adjacent (incident) to a vertex (edge) in S and to a vertex (edge) in V - S(E-S). The minimum cardinality of a restrained dominating (restrained edge dominating) set of G is called restrained domination (restrained edge domination) number of G, denoted by  γr (G) (γre(G). The restrained edge domination number of some standard graphs are already investigated while in this paper the restrained edge domination number like degree splitting, switching,  square and middle graph obtained from path.

Independent domination in signed graphs

2020 ◽  
pp. 2150065
Author(s):  
P. Jeyalakshmi ◽  
K. Karuppasamy ◽  
S. Arockiaraj
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Signed Graph ◽  
Signed Graphs ◽  
Independent Domination ◽  
Independent Dominating Set

Let [Formula: see text] be a signed graph. A dominating set [Formula: see text] is said to be an independent dominating set of [Formula: see text] if [Formula: see text] is a fully negative. In this paper, we initiate a study of this parameter. We also establish the bounds and characterization on the independent domination number of a signed graph.

scholarly journals -Tuple Total Restrained Domination in Complementary Prisms

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Adel P. Kazemi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Restrained Domination ◽  
Minimum Number ◽  
Total Restrained Domination ◽  
Complementary Prism ◽  
Total Restrained Domination Number ◽  
Restrained Domination Number

In a graph with , a -tuple total restrained dominating set is a subset of such that each vertex of   is adjacent to at least vertices of and also each vertex of is adjacent to at least vertices of  . The minimum number of vertices of such sets in is the -tuple total restrained domination number of . In [-tuple total restrained domination/domatic in graphs, BIMS], the author initiated the study of the -tuple total restrained domination number in graphs. In this paper, we continue it in the complementary prism of a graph.

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.

scholarly journals On the Aα-Eigenvalues of Signed Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1990
Author(s):  
Germain Pastén ◽  
Oscar Rojo ◽  
Luis Medina
Keyword(s):  
Adjacency Matrix ◽  
Undirected Graph ◽  
Upper Bounds ◽  
Signed Graph ◽  
Lower And Upper Bounds ◽  
Signed Graphs ◽  
Basic Properties ◽  
Underlying Graph ◽  
Positive Semidefiniteness ◽  
Vertex Degrees

For α∈[0,1], let Aα(Gσ)=αD(G)+(1−α)A(Gσ), where G is a simple undirected graph, D(G) is the diagonal matrix of its vertex degrees and A(Gσ) is the adjacency matrix of the signed graph Gσ whose underlying graph is G. In this paper, basic properties of Aα(Gσ) are obtained, its positive semidefiniteness is studied and some bounds on its eigenvalues are derived—in particular, lower and upper bounds on its largest eigenvalue are obtained.

scholarly journals Resolving Restrained Domination in Graphs

2021 ◽  
Vol 14 (3) ◽  
pp. 829-841
Author(s):  
Gerald Bacon Monsanto ◽  
Helen M. Rara
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Restrained Domination ◽  
Isolated Vertex ◽  
Domination In Graphs ◽  
Product Of Graphs ◽  
Restrained Domination Number

Let G be a connected graph. Brigham et al. [3] defined a resolving dominating setas a set S of vertices of a connected graph G that is both resolving and dominating. A set S ⊆ V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or hV (G) \ Si has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these graphs.

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