Tadpole domination in duplicated graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830921500038 ◽

2020 ◽

pp. 2150003

Author(s):

M. N. Al-Harere ◽

P. A. Khuda Bakhash

Keyword(s):

Domination Number ◽

Hamiltonian Path ◽

Original Graph ◽

Maximal Path ◽

Initial Graph

In this paper, selected domination parameters are discussed and proved especially after expanding the graph by duplicating vertices or edges. The tadpole domination number after expansion is obtained in terms of the old tadpole domination number and the maximal path of the original graph, tadpole domination number was determined for any graph after the duplication of the order of the vertices set of G or the duplication of the size of the edges set. Determining if a graph is Hamiltonian is much more problematic. Therefore, the leading outcome in this paper is that we provide that if an expanded graph (by duplicating each vertex by an edge) has tadpole domination, then the original (initial) graph has Hamiltonian path and vice versa.

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BOUNDS ON THE DOMINATION NUMBER OF PERMUTATION GRAPHS

Journal of Interconnection Networks ◽

10.1142/s0219265909002522 ◽

2009 ◽

Vol 10 (03) ◽

pp. 205-217 ◽

Cited By ~ 2

Author(s):

WEIZHEN GU ◽

KIRSTI WASH

Keyword(s):

Maximum Degree ◽

Domination Number ◽

Permutation Graph ◽

Original Graph ◽

Permutation Graphs ◽

Positive Integers

For a graph G with n vertices and a permutation α on V(G), a permutation graph Pα(G) is obtained from two identical copies of G by adding an edge between v and α(V) for any v ϵ V(G). Let γ(G) be the domination number of a graph G. It has been shown that γ(G) ≤ γ(Pα(G) ≤ 2γ(G) for any permutation α on V(G). In this paper, we investigate specific graphs for which there exists a permutation α such that γ(Pα(G)) ≻ γ(G) in terms of the domination number of G or the maximum degree of G. Additionally, we construct a class of graphs for which the domination number of any permutation graph is twice the domination number of the original graph, as well as explore finding a specific graph G and permutation α for any two positive integers a and b with a ≤ b ≤ 2a, to have γ(G) = a and γ(Pα(G)) = b.

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Total Domination on Some Graph Operators

Mathematics ◽

10.3390/math9030241 ◽

2021 ◽

Vol 9 (3) ◽

pp. 241

Author(s):

José M. Sigarreta

Keyword(s):

Dominating Set ◽

Domination Number ◽

Total Domination ◽

Dominating Sets ◽

Total Domination Number ◽

Arbitrary Graph ◽

Original Graph ◽

Minimum Cardinality ◽

Total Dominating Set

Let G=(V,E) be a graph; a set D⊆V is a total dominating set if every vertex v∈V has, at least, one neighbor in D. The total domination number γt(G) is the minimum cardinality among all total dominating sets. Given an arbitrary graph G, we consider some operators on this graph; S(G),R(G), and Q(G), and we give bounds or the exact value of the total domination number of these new graphs using some parameters in the original graph G.

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Medium Domination Number of Certain Graphs

International Journal of Emerging Research in Management and Technology ◽

10.23956/ijermt.v6i11.55 ◽

2018 ◽

Vol 6 (11) ◽

pp. 130

Author(s):

K. Uma Samundesvari ◽

J. Maria Regila Baby

Keyword(s):

Domination Number

In this paper the author have found out the medium domination number of Helm graph, Friendship graph.

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Roulette Wheel Selection based Heuristic Algorithm for the Orienteering Problem

INTERNATIONAL JOURNAL OF COMPUTERS & TECHNOLOGY ◽

10.24297/ijct.v13i1.2933 ◽

2014 ◽

Vol 13 (1) ◽

pp. 4127-4145

Author(s):

Madhushi Verma ◽

Mukul Gupta ◽

Bijeeta Pal ◽

Prof. K. K. Shukla

Keyword(s):

Triangle Inequality ◽

Time Budget ◽

Control Point ◽

Hamiltonian Path ◽

Real Life ◽

Tourism Industry ◽

Orienteering Problem ◽

Complete Graphs ◽

Roulette Wheel Selection ◽

Roulette Wheel

Orienteering problem (OP) is an NP-Hard graph problem. The nodes of the graph are associated with scores or rewards and the edges with time delays. The goal is to obtain a Hamiltonian path connecting the two necessary check points, i.e. the source and the target along with a set of control points such that the total collected score is maximized within a specified time limit. OP finds application in several fields like logistics, transportation networks, tourism industry, etc. Most of the existing algorithms for OP can only be applied on complete graphs that satisfy the triangle inequality. Real-life scenario does not guarantee that there exists a direct link between all control point pairs or the triangle inequality is satisfied. To provide a more practical solution, we propose a stochastic greedy algorithm (RWS_OP) that uses the roulette wheel selectionmethod, does not require that the triangle inequality condition is satisfied and is capable of handling both complete as well as incomplete graphs. Based on several experiments on standard benchmark data we show that RWS_OP is faster, more efficient in terms of time budget utilization and achieves a better performance in terms of the total collected score ascompared to a recently reported algorithm for incomplete graphs.

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