scholarly journals Tight Upper Bounds for the Domination Numbers of Graphs with Given Order and Minimum Degree

10.37236/1311 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
W. Edwin Clark ◽  
Larry A. Dunning
Keyword(s):  
Minimum Degree ◽  
Domination Number ◽  
Upper Bounds ◽  
Domination Numbers

Let $\gamma(n,\delta)$ denote the maximum possible domination number of a graph with $n$ vertices and minimum degree $\delta$. Using known results we determine $\gamma(n,\delta)$ for $\delta = 0, 1, 2, 3$, $n \ge \delta + 1$ and for all $n$, $\delta$ where $\delta = n-k$ and $n$ is sufficiently large relative to $k$. We also obtain $\gamma(n,\delta)$ for all remaining values of $(n,\delta)$ when $n \le 14$ and all but 6 values of $(n,\delta)$ when $n = 15$ or 16.

scholarly journals Tight Upper Bounds for the Domination Numbers of Graphs with Given Order and Minimum Degree, II

10.37236/1536 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
W. Edwin Clark ◽  
Larry A. Dunning ◽  
Stephen Suen
Keyword(s):  
Lower Bound ◽  
Minimum Degree ◽  
Domination Number ◽  
Difference Set ◽  
Upper Bounds ◽  
Circulant Graphs ◽  
Large N ◽  
The Right ◽  
Domination Numbers

Let $\gamma (n,\delta)$ denote the largest possible domination number for a graph of order $n$ and minimum degree $\delta$. This paper is concerned with the behavior of the right side of the sequence $$\gamma (n,0) \ge \gamma (n,1) \ge \cdots \ge \gamma (n,n-1) = 1. $$ We set $ \delta _k(n) = \max \{ \delta \, \vert \, \gamma (n,\delta) \ge k \}$, $k \ge 1.$ Our main result is that for any fixed $k \ge 2$ there is a constant $c_k$ such that for sufficiently large $n$, $$ n-c_kn^{(k-1)/k} \le \delta _{k+1}(n) \le n - n^{(k-1)/k}. $$ The lower bound is obtained by use of circulant graphs. We also show that for $n$ sufficiently large relative to $k$, $\gamma (n,\delta _k(n)) = k$. The case $k=3$ is examined in further detail. The existence of circulant graphs with domination number greater than 2 is related to a kind of difference set in ${\bf Z}_n$.

scholarly journals Graphs with Constant Sum of Domination and Inverse Domination Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
T. Asir
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Complete Characterization ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Vertex Set ◽  
Domination Numbers

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

scholarly journals On 3-Rainbow Domination Number of Generalized Petersen Graphs P(6k,k)

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1860
Author(s):  
Rija Erveš ◽  
Janez Žerovnik
Keyword(s):  
Domination Number ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Generalized Petersen Graphs ◽  
Rainbow Domination ◽  
Petersen Graphs ◽  
Special Case ◽  
Domination Numbers

We obtain new results on 3-rainbow domination numbers of generalized Petersen graphs P(6k,k). In some cases, for some infinite families, exact values are established; in all other cases, the lower and upper bounds with small gaps are given. We also define singleton rainbow domination, where the sets assigned have a cardinality of, at most, one, and provide analogous results for this special case of rainbow domination.

On Double Domination Numbers of Generalized Petersen Graphs

2016 ◽  
Vol 13 (10) ◽  
pp. 6514-6518
Author(s):  
Minhong Sun ◽  
Zehui Shao
Keyword(s):  
Programming Model ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Minimum Size ◽  
Generalized Petersen Graphs ◽  
Small Parameters ◽  
Petersen Graphs ◽  
Double Domination ◽  
Domination Numbers

A (total) double dominating set in a graph G is a subset S ⊆ V(G) such that each vertex in V(G) is (total) dominated by at least 2 vertices in S. The (total) double domination number of G is the minimum size of a (total) double dominating set of G. We determine the total double domination numbers and give upper bounds for double domination numbers of generalized Petersen graphs. By applying an integer programming model for double domination numbers of a graph, we have determined some exact values of double domination numbers of these generalized Petersen graphs with small parameters. The result shows that the given upper bounds match these exact values.

Signed domination and Mycielski’s structure in graphs

2020 ◽  
Vol 54 (4) ◽  
pp. 1077-1086
Author(s):  
Arezoo N. Ghameshlou ◽  
Athena Shaminezhad ◽  
Ebrahim Vatandoost ◽  
Abdollah Khodkar
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Minimum Weight ◽  
Domination Number ◽  
Upper Bounds ◽  
Underlying Graph ◽  
Signed Domination ◽  
Signed Domination Number ◽  
Dominating Function ◽  
Domination Numbers

Let G = (V, E) be a graph. The function f : V(G) → {−1, 1} is a signed dominating function if for every vertex v ∈ V(G), ∑x∈NG[v] f(x)≥1. The value of ω(f) = ∑x∈V(G) f(x) is called the weight of f. The signed domination number of G is the minimum weight of a signed dominating function of G. In this paper, we initiate the study of the signed domination numbers of Mycielski graphs and find some upper bounds for this parameter. We also calculate the signed domination number of the Mycielski graph when the underlying graph is a star, a wheel, a fan, a Dutch windmill, a cycle, a path or a complete bipartite graph.

scholarly journals Domination number within on-line social networks

2021 ◽  
Author(s):  
Marc Lozier
Keyword(s):  
Social Networks ◽  
Domination Number ◽  
Upper Bounds ◽  
Dominating Sets ◽  
Geometric Distance ◽  
New Model ◽  
On Line ◽  
Domination Numbers

There is particular interest in on-line social networks (OSNs) and capturing their properties. The memoryless geometric protean (MGEO-P) model provably simulated many OSN properties. We investigated dominating sets in OSNs and their models. The domination numbers were computed using two algorithms, DS-DC and DS-RAI, for MGEO-P samples and Facebook data, known as the Facebook 100 graphs. We establish sub-linear bounds on the domination numbers for the Facebook 100 graphs, and show that these bounds correlate well with bounds in graphs simulated by MGEO-P. A new model is introduced known as the Distance MGEO-P (DMGEO-P) model. This model incorporates geometric distance to inuence the probability that two nodes are adjacent. Domination number upper bounds were found to be well-correlated with the Facebook 100 graph.

scholarly journals Domination number within on-line social networks

2021 ◽  
Author(s):  
Marc Lozier
Keyword(s):  
Social Networks ◽  
Domination Number ◽  
Upper Bounds ◽  
Dominating Sets ◽  
Geometric Distance ◽  
New Model ◽  
On Line ◽  
Domination Numbers

There is particular interest in on-line social networks (OSNs) and capturing their properties. The memoryless geometric protean (MGEO-P) model provably simulated many OSN properties. We investigated dominating sets in OSNs and their models. The domination numbers were computed using two algorithms, DS-DC and DS-RAI, for MGEO-P samples and Facebook data, known as the Facebook 100 graphs. We establish sub-linear bounds on the domination numbers for the Facebook 100 graphs, and show that these bounds correlate well with bounds in graphs simulated by MGEO-P. A new model is introduced known as the Distance MGEO-P (DMGEO-P) model. This model incorporates geometric distance to inuence the probability that two nodes are adjacent. Domination number upper bounds were found to be well-correlated with the Facebook 100 graph.

scholarly journals Private Edge Domination Number of a Graph

2013 ◽  
Vol 5 (2) ◽  
pp. 283-294
Author(s):  
Kavitha S ◽  
Robinson C. S
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Dominating Sets ◽  
Edge Dominating Set ◽  
Domination Numbers

A set    is said to be a private edge dominating set, if it is an edge dominating set, for every has at least one external private neighbor in . Let  and  denote the minimum and maximum cardinalities, respectively, of a private edge dominating sets in a graph . In this paper we characterize connected graph for which ? q/2 and the graph for some upper bounds. The private edge domination numbers of several classes of graphs are determined.Keywords: Edge domination; Perfect domination; Private domination; Edge irredundant sets.© 2013 JSR Publications. ISSN: 2070-0237 (Print); 2070-0245 (Online). All rights reserved.doi: http://dx.doi.org/10.3329/jsr.v5i2.12024         J. Sci. Res. 5 (2), 283-294 (2013)

scholarly journals Upper bounds on the signed edge domination number of a graph

2021 ◽  
Vol 344 (2) ◽  
pp. 112201
Author(s):  
Fengming Dong ◽  
Jun Ge ◽  
Yan Yang
Keyword(s):  
Domination Number ◽  
Upper Bounds ◽  
Signed Edge Domination Number
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