scholarly journals Heredity for generalized power domination

2016 ◽  
Vol Vol. 18 no. 3 (Graph Theory) ◽  
Author(s):  
Paul Dorbec ◽  
Seethu Varghese ◽  
Ambat Vijayakumar
Keyword(s):  
Domination Number ◽  
Power Domination ◽  
Edge Contraction ◽  
Vertex Deletion ◽  
International Audience ◽  
Optimal Bounds

International audience In this paper, we study the behaviour of the generalized power domination number of a graph by small changes on the graph, namely edge and vertex deletion and edge contraction. We prove optimal bounds for $\gamma_{p,k}(G-e)$, $\gamma_{p,k}(G/e)$ and for $\gamma_{p,k}(G-v)$ in terms of $\gamma_{p,k}(G)$, and give examples for which these bounds are tight. We characterize all graphs for which $\gamma_{p,k}(G-e) = \gamma_{p,k}(G)+1$ for any edge $e$. We also consider the behaviour of the propagation radius of graphs by similar modifications.

scholarly journals Study on power domination number of some classes of graphs

2020 ◽  
Author(s):  
T. N. Saibavani ◽  
N. Parvathi
Keyword(s):  
Domination Number ◽  
Power Domination

scholarly journals Equitable power domination number of total graph of certain graphs

2020 ◽  
Vol 1531 ◽  
pp. 012073
Author(s):  
S Banu Priya ◽  
A Parthiban ◽  
P Abirami
Keyword(s):  
Domination Number ◽  
Total Graph ◽  
Power Domination

Power domination number of sunlet graph and other graphs

2019 ◽  
Vol 22 (6) ◽  
pp. 1121-1127
Author(s):  
T. N. Saibavani ◽  
N. Parvathi
Keyword(s):  
Domination Number ◽  
Power Domination

scholarly journals Edge stability in secure graph domination

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Anton Pierre Burger ◽  
Alewyn Petrus Villiers ◽  
Jan Harm Vuuren
Keyword(s):  
Graph Theory ◽  
Dominating Set ◽  
Domination Number ◽  
Arbitrary Subset ◽  
Graph Domination ◽  
Vertex Set ◽  
International Audience ◽  
Edge Stability ◽  
Secure Domination

Graph Theory International audience A subset X of the vertex set of a graph G is a secure dominating set of G if X is a dominating set of G and if, for each vertex u not in X, there is a neighbouring vertex v of u in X such that the swap set (X-v)∪u is again a dominating set of G. The secure domination number of G is the cardinality of a smallest secure dominating set of G. A graph G is p-stable if the largest arbitrary subset of edges whose removal from G does not increase the secure domination number of the resulting graph, has cardinality p. In this paper we study the problem of computing p-stable graphs for all admissible values of p and determine the exact values of p for which members of various infinite classes of graphs are p-stable. We also consider the problem of determining analytically the largest value ωn of p for which a graph of order n can be p-stable. We conjecture that ωn=n-2 and motivate this conjecture.

scholarly journals Guarded subgraphs and the domination game

2015 ◽  
Vol Vol. 17 no. 1 (Graph Theory) ◽  
Author(s):  
Boštjan Brešar ◽  
Sandi Klavžar ◽  
Gasper Košmrlj ◽  
Doug F. Rall
Keyword(s):  
Graph Theory ◽  
Domination Number ◽  
Metric Properties ◽  
International Audience ◽  
Game Domination Number ◽  
Domination Game

Graph Theory International audience We introduce the concept of guarded subgraph of a graph, which as a condition lies between convex and 2-isometric subgraphs and is not comparable to isometric subgraphs. Some basic metric properties of guarded subgraphs are obtained, and then this concept is applied to the domination game. In this game two players, Dominator and Staller, alternate choosing vertices of a graph, one at a time, such that each chosen vertex enlarges the set of vertices dominated so far. The aim of Dominator is that the graph is dominated in as few steps as possible, while the aim of Staller is just the opposite. The game domination number is the number of vertices chosen when Dominator starts the game and both players play optimally. The main result of this paper is that the game domination number of a graph is not smaller than the game domination number of any guarded subgraph. Several applications of this result are presented.

scholarly journals Total domination in K₅- and K₆-covered graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Odile Favaron ◽  
H. Karami ◽  
S. M. Sheikholeslami
Keyword(s):  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
International Audience

Graphs and Algorithms International audience A graph G is Kr-covered if each vertex of G is contained in a Kr-clique. Let $\gamma_t(G)$ denote the total domination number of G. It has been conjectured that every Kr-covered graph of order n with no Kr-component satisfies $\gamma_t(G) \le \frac{2n}{r+1}$. We prove that this conjecture is true for r = 5 and 6.

scholarly journals On the Power Domination Number of de Bruijn and Kautz Digraphs

2018 ◽  
pp. 264-272 ◽  
Author(s):  
Cyriac Grigorious ◽  
Thomas Kalinowski ◽  
Sudeep Stephen
Keyword(s):  
Domination Number ◽  
Power Domination ◽  
Kautz Digraphs ◽  
De Bruijn

scholarly journals Note on power propagation time and lower bounds for the power domination number

2016 ◽  
Vol 34 (3) ◽  
pp. 736-741 ◽  
Author(s):  
Daniela Ferrero ◽  
Leslie Hogben ◽  
Franklin H. J. Kenter ◽  
Michael Young
Keyword(s):  
Lower Bounds ◽  
Domination Number ◽  
Propagation Time ◽  
Power Domination

scholarly journals Forcing Subsets for γ∗ tpw -sets in Graphs

2021 ◽  
Vol 14 (2) ◽  
pp. 451-470
Author(s):  
Cris Laquibla Armada
Keyword(s):  
Domination Number ◽  
Upper Bounds ◽  
Complete Graphs ◽  
Lower And Upper Bounds ◽  
Power Domination ◽  
Wheel Graphs

In this paper, the lower and upper bounds of the forcing total dr-power dominationnumber of any graph are determined. Total dr-power domination number of some special graphs such as complete graphs, star, fan and wheel graphs are shown. Moreover, the forcing total dr-power domination number of these graphs, together with paths and cycles, are determined.

scholarly journals Power Domination Number On Shackle Operation with Points as Lingkage

2020 ◽  
Vol 4 (1) ◽  
pp. 1
Author(s):  
Ilham Saifudin
Keyword(s):  
Mathematical Logic ◽  
Maximum Degree ◽  
Nearest Neighbor ◽  
Dominating Set ◽  
Domination Number ◽  
Axiomatic Method ◽  
Star Graph ◽  
Power Domination ◽  
Path Graph ◽  
Deductive Method

The Power dominating set is a minimum point of determination in a graph that can dominate the connected dots around it, with a minimum domination point. The smallest cardinality of a power dominating set is called a power domination number with the notation . The purpose of this study is to determine the Shackle operations graph value from several special graphs with a point as a link. The result operation graphs are: Shackle operation graph from Path graph , Shackle operation graph from Sikel graph , Shackle operation graph from Star graph . The method used in this paper is axiomatic deductive method in solving problems. Understanding the axiomatic method itself is a method of deductive proof principles that applies in mathematical logic by using theorems that already exist in solving a problem. In this paper begins by determining the paper object that is the Shackle point operations result graph. Next, determine the cardinality of these graphs. After that, determine the point that has the maximum degree on the graph as the dominator point of power domination. Then, check whether the nearest neighbor has two or more degrees and analyze its optimization by using a ceiling function comparison between zero forching with the greatest degree of graph. Thus it can be determined ϒp minimal and dominated. The results of the power domination number study on Shackle operation graph result with points as connectors are , for  and ; , for  and ; , for  and .

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