scholarly journals [1,2]-Domination in generalized Petersen graphs

2019 ◽  
Vol 11 (05) ◽  
pp. 1950058
Author(s):  
Fairouz Beggas ◽  
Volker Turau ◽  
Mohammed Haddad ◽  
Hamamache Kheddouci
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Generalized Petersen Graphs ◽  
Petersen Graphs ◽  
Domination Numbers

A vertex subset [Formula: see text] of a graph [Formula: see text] is a [Formula: see text]-dominating set if each vertex of [Formula: see text] is adjacent to either one or two vertices in [Formula: see text]. The minimum cardinality of a [Formula: see text]-dominating set of [Formula: see text], denoted by [Formula: see text], is called the [Formula: see text]-domination number of [Formula: see text]. In this paper, the [Formula: see text]-domination and the [Formula: see text]-total domination numbers of the generalized Petersen graphs [Formula: see text] are determined.

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.

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.

scholarly journals The Numerical Invariants concerning the Total Domination for Generalized Petersen Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Taiyin Zhao ◽  
Gohar Ali ◽  
Nabila Hameed ◽  
Syed Inayat Ali Shah ◽  
Yu-Ming Chu
Keyword(s):  
Dominating Set ◽  
Total Domination ◽  
Dominating Sets ◽  
Maximum Order ◽  
Minimum Cardinality ◽  
Generalized Petersen Graphs ◽  
Total Dominating Set ◽  
Domatic Number ◽  
Petersen Graphs ◽  
Numerical Invariants

A subset S of V G is called a total dominating set of a graph G if every vertex in V G is adjacent to a vertex in S . The total domination number of a graph G denoted by γ t G is the minimum cardinality of a total dominating set in G . The maximum order of a partition of V G into total dominating sets of G is called the total domatic number of G and is denoted by d t G . Domination in graphs has applications to several fields. Domination arises in facility location problems, where the number of facilities (e.g., hospitals and fire stations) is fixed, and one attempts to minimize the distance that a person needs to travel to get to the closest facility. In this paper, the numerical invariants concerning the total domination are studied for generalized Petersen graphs.

scholarly journals Trees with equal total domination and 2-rainbow domination numbers

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 599-607 ◽  
Author(s):  
Zehui Shao ◽  
Seyed Sheikholeslamib ◽  
Bo Wang ◽  
Pu Wu ◽  
Xiaosong Zhang
Keyword(s):  
Dominating Set ◽  
Minimum Weight ◽  
Domination Number ◽  
Total Domination ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Rainbow Domination ◽  
Domination In Graphs ◽  
Appl Math ◽  
Domination Numbers

A 2-rainbow dominating function (2RDF) of a graph G is a function f : V(G) ? P({1,2}) such that for each v ? V(G) with f (v) = ? we have Uu?N(v) f (u) = {1,2}. For a 2RDF f of a graph G, the weight w(f) of f is defined as w(f)=?v?V(G)?f(v)?. The minimum weight over all 2RDFs of G is called the 2-rainbow domination number of G, which is denoted by ?r2(G). A subset S of vertices of a graph G without isolated vertices, is a total dominating set of G if every vertex in V(G) has a neighbor in S. The total domination number ?t(G) is the minimum cardinality of a total dominating set of G. Chellali, Haynes and Hedetniemi conjectured that ?t(G)? ?r2(G) [M. Chellali, T.W. Haynes and S.T. Hedetniemi, Bounds on weak Roman and 2-rainbow domination numbers, Discrete Appl. Math. 178 (2014), 27-32.], and later Furuya confirmed the conjecture [M. Furuya, A note on total domination and 2-rainbow domination in graphs, Discrete Appl. Math. 184 (2015), 229-230.]. In this paper, we provide a constructive characterization of trees T with ?r2(T) = ?t(T).

A note on improved upper bounds on the transversal number of hypergraphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950004
Author(s):  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Size ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

scholarly journals New Bounds on the Double Total Domination Number of Graphs

2021 ◽  
Author(s):  
A. Cabrera-Martínez ◽  
F. A. Hernández-Mira
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Dominating Sets ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set

AbstractLet G be a graph of minimum degree at least two. A set $$D\subseteq V(G)$$ D ⊆ V ( G ) is said to be a double total dominating set of G if $$|N(v)\cap D|\ge 2$$ | N ( v ) ∩ D | ≥ 2 for every vertex $$v\in V(G)$$ v ∈ V ( G ) . The minimum cardinality among all double total dominating sets of G is the double total domination number of G. In this article, we continue with the study of this parameter. In particular, we provide new bounds on the double total domination number in terms of other domination parameters. Some of our results are tight bounds that improve some well-known results.

Weak and strong domination in thorn graphs

2019 ◽  
Vol 13 (04) ◽  
pp. 2050071
Author(s):  
Derya Doğan Durgun ◽  
Berna Lökçü
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Thorn Graphs ◽  
Domination Numbers

Let [Formula: see text] be a graph and [Formula: see text] A dominating set [Formula: see text] is a set of vertices such that each vertex of [Formula: see text] is either in [Formula: see text] or has at least one neighbor in [Formula: see text]. The minimum cardinality of such a set is called the domination number of [Formula: see text], [Formula: see text] [Formula: see text] strongly dominates [Formula: see text] and [Formula: see text] weakly dominates [Formula: see text] if (i) [Formula: see text] and (ii) [Formula: see text] A set [Formula: see text] is a strong-dominating set, shortly sd-set, (weak-dominating set, shortly wd-set) of [Formula: see text] if every vertex in [Formula: see text] is strongly (weakly) dominated by at least one vertex in [Formula: see text]. The strong (weak) domination number [Formula: see text] of [Formula: see text] is the minimum cardinality of an sd-set (wd-set). In this paper, we present weak and strong domination numbers of thorn graphs.

Disjunctive total domination in permutation graphs

2017 ◽  
Vol 09 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Eunjeong Yi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Number Formula

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. If [Formula: see text] has no isolated vertex, then a disjunctive total dominating set (DTD-set) of [Formula: see text] is a vertex set [Formula: see text] such that every vertex in [Formula: see text] is adjacent to a vertex of [Formula: see text] or has at least two vertices in [Formula: see text] at distance two from it, and the disjunctive total domination number [Formula: see text] of [Formula: see text] is the minimum cardinality overall DTD-sets of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two disjoint copies of a graph [Formula: see text], and let [Formula: see text] be a bijection. Then, a permutation graph [Formula: see text] has the vertex set [Formula: see text] and the edge set [Formula: see text]. For any connected graph [Formula: see text] of order at least three, we prove the sharp bounds [Formula: see text]; we give an example showing that [Formula: see text] can be arbitrarily large. We characterize permutation graphs for which [Formula: see text] holds. Further, we show that [Formula: see text] when [Formula: see text] is a cycle, a path, and a complete [Formula: see text]-partite graph, respectively.

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.

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