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 Independent Domination Stable Trees and Unicyclic Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 820
Author(s):  
Pu Wu ◽  
Huiqin Jiang ◽  
Sakineh Nazari-Moghaddam ◽  
Seyed Mahmoud Sheikholeslami ◽  
Zehui Shao ◽  
...  
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Unicyclic Graphs ◽  
Basic Properties ◽  
Independent Domination ◽  
Independent Dominating Set

A set S ⊆ V ( G ) in a graph G is a dominating set if every vertex of G is either in S or adjacent to a vertex of S . A dominating set S is independent if any pair of vertices in S is not adjacent. The minimum cardinality of an independent dominating set on a graph G is called the independent domination number i ( G ) . A graph G is independent domination stable if the independent domination number of G remains unchanged under the removal of any vertex. In this paper, we study the basic properties of independent domination stable graphs, and we characterize all independent domination stable trees and unicyclic graphs. In addition, we establish bounds on the order of independent domination stable trees.

scholarly journals On the Total Outer k-Independent Domination Number of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 194 ◽  
Author(s):  
Abel Cabrera-Martínez ◽  
Juan Carlos Hernández-Gómez ◽  
Ernesto Parra-Inza ◽  
José María Sigarreta Almira
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Np Hard Problem ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Computational Properties

A set of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex in such a set. We say that a total dominating set D is a total outer k-independent dominating set of G if the maximum degree of the subgraph induced by the vertices that are not in D is less or equal to k − 1 . The minimum cardinality among all total outer k-independent dominating sets is the total outer k-independent domination number of G. In this article, we introduce this parameter and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this novel parameter and other ones related to domination and independence in graphs. In addition, we give several Nordhaus–Gaddum type results. Finally, we prove that computing the total outer k-independent domination number of a graph G is an NP-hard problem.

Independent domination number in Cayley digraphs of rectangular groups

2018 ◽  
Vol 10 (02) ◽  
pp. 1850024
Author(s):  
Nuttawoot Nupo ◽  
Sayan Panma
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Left Zero ◽  
Zero Semigroup ◽  
Left Zero Semigroup ◽  
Group A ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Rectangular Group

Let [Formula: see text] denote the Cayley digraph of the rectangular group [Formula: see text] with respect to the connection set [Formula: see text] in which the rectangular group [Formula: see text] is isomorphic to the direct product of a group, a left zero semigroup, and a right zero semigroup. An independent dominating set of [Formula: see text] is the independent set of elements in [Formula: see text] that can dominate the whole elements. In this paper, we investigate the independent domination number of [Formula: see text] and give more results on Cayley digraphs of left groups and right groups which are specific cases of rectangular groups. Moreover, some results of the path independent domination number of [Formula: see text] are also shown.

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.

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 Common Independence in Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1411
Author(s):  
Magda Dettlaff ◽  
Magdalena Lemańska ◽  
Jerzy Topp
Keyword(s):  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Independent Subset ◽  
Block Graphs ◽  
Independent Domination ◽  
The Common ◽  
Independent Dominating Set

The cardinality of a largest independent set of G, denoted by α(G), is called the independence number of G. The independent domination number i(G) of a graph G is the cardinality of a smallest independent dominating set of G. We introduce the concept of the common independence number of a graph G, denoted by αc(G), as the greatest integer r such that every vertex of G belongs to some independent subset X of VG with |X|≥r. The common independence number αc(G) of G is the limit of symmetry in G with respect to the fact that each vertex of G belongs to an independent set of cardinality αc(G) in G, and there are vertices in G that do not belong to any larger independent set in G. For any graph G, the relations between above parameters are given by the chain of inequalities i(G)≤αc(G)≤α(G). In this paper, we characterize the trees T for which i(T)=αc(T), and the block graphs G for which αc(G)=α(G).

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.

scholarly journals Independent Domination Subdivision in Graphs

2021 ◽  
Author(s):  
Ammar Babikir ◽  
Magda Dettlaff ◽  
Michael A. Henning ◽  
Magdalena Lemańska
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Set ◽  
Minimum Cardinality ◽  
Minimum Number ◽  
Independent Domination ◽  
Delta G ◽  
Independent Dominating Set ◽  
Domination Subdivision Number

AbstractA set S of vertices in a graph G is a dominating set if every vertex not in S is ad jacent to a vertex in S. If, in addition, S is an independent set, then S is an independent dominating set. The independent domination number i(G) of G is the minimum cardinality of an independent dominating set in G. The independent domination subdivision number $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( 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 independent domination number. We show that for every connected graph G on at least three vertices, the parameter $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) is well defined and differs significantly from the well-studied domination subdivision number $$\mathrm{sd_\gamma }(G)$$ sd γ ( G ) . For example, if G is a block graph, then $$\mathrm{sd_\gamma }(G) \le 3$$ sd γ ( G ) ≤ 3 , while $$ \hbox {sd}_{\mathrm{i}}(G)$$ sd i ( G ) can be arbitrary large. Further we show that there exist connected graph G with arbitrarily large maximum degree $$\Delta (G)$$ Δ ( G ) such that $$ \hbox {sd}_{\mathrm{i}}(G) \ge 3 \Delta (G) - 2$$ sd i ( G ) ≥ 3 Δ ( G ) - 2 , in contrast to the known result that $$\mathrm{sd_\gamma }(G) \le 2 \Delta (G) - 1$$ sd γ ( G ) ≤ 2 Δ ( G ) - 1 always holds. Among other results, we present a simple characterization of trees T with $$ \hbox {sd}_{\mathrm{i}}(T) = 1$$ sd i ( T ) = 1 .

scholarly journals A lower bound on the double outer-independent domination number of a tree

2012 ◽  
Vol 45 (1) ◽  
Author(s):  
Marcin Krzywkowski
Keyword(s):  
Lower Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Independent Domination ◽  
Independent Dominating Set

AbstractA vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph

scholarly journals Independent Domination Number in Adaptive Mesh Refinement (AMR)-WENO Scheme Networks

2020 ◽  
Vol 8 (4S5) ◽  
pp. 17-19
Keyword(s):  
Adaptive Mesh Refinement ◽  
Dominating Set ◽  
Domination Number ◽  
Mesh Refinement ◽  
Adaptive Mesh ◽  
Weno Scheme ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Independent Domination ◽  
Independent Dominating Set

Let G be the graph, consider the vertex set as V and edge set as E. If S is the subset of the vertex set V such that S contains vertices which has atleast one neighbor in V that is not in S, then S is said to be dominating set of G. If the vertex in S is not adjacent to one another, then S is called as the independent dominating set of G and so i(G) represents the independent domination number, the minimum cardinality of an independent dominating set in G. In this paper, we obtain independent domination number for triangular, quadrilateral, pentagonal, hexagonal, heptagonal and octagonal networks by Adaptive Mesh Refinement (AMR)-WENO Scheme.

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