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 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 Distance Domination and Distance Irredundance in Graphs

10.37236/953 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Adriana Hansberg ◽  
Dirk Meierling ◽  
Lutz Volkmann
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Distance Domination

A set $D\subseteq V$ of vertices is said to be a (connected) distance $k$-dominating set of $G$ if the distance between each vertex $u\in V-D$ and $D$ is at most $k$ (and $D$ induces a connected graph in $G$). The minimum cardinality of a (connected) distance $k$-dominating set in $G$ is the (connected) distance $k$-domination number of $G$, denoted by $\gamma_k(G)$ ($\gamma_k^c(G)$, respectively). The set $D$ is defined to be a total $k$-dominating set of $G$ if every vertex in $V$ is within distance $k$ from some vertex of $D$ other than itself. The minimum cardinality among all total $k$-dominating sets of $G$ is called the total $k$-domination number of $G$ and is denoted by $\gamma_k^t(G)$. For $x\in X\subseteq V$, if $N^k[x]-N^k[X-x]\neq\emptyset$, the vertex $x$ is said to be $k$-irredundant in $X$. A set $X$ containing only $k$-irredundant vertices is called $k$-irredundant. The $k$-irredundance number of $G$, denoted by $ir_k(G)$, is the minimum cardinality taken over all maximal $k$-irredundant sets of vertices of $G$. In this paper we establish lower bounds for the distance $k$-irredundance number of graphs and trees. More precisely, we prove that ${5k+1\over 2}ir_k(G)\geq \gamma_k^c(G)+2k$ for each connected graph $G$ and $(2k+1)ir_k(T)\geq\gamma_k^c(T)+2k\geq |V|+2k-kn_1(T)$ for each tree $T=(V,E)$ with $n_1(T)$ leaves. A class of examples shows that the latter bound is sharp. The second inequality generalizes a result of Meierling and Volkmann and Cyman, Lemańska and Raczek regarding $\gamma_k$ and the first generalizes a result of Favaron and Kratsch regarding $ir_1$. Furthermore, we shall show that $\gamma_k^c(G)\leq{3k+1\over2}\gamma_k^t(G)-2k$ for each connected graph $G$, thereby generalizing a result of Favaron and Kratsch regarding $k=1$.

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 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.

On the domination number of a graph and its total graph

2020 ◽  
Vol 12 (05) ◽  
pp. 2050068
Author(s):  
E. Murugan ◽  
J. Paulraj Joseph
Keyword(s):  
Lower Bounds ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Total Graph

In this paper, we investigate the upper and lower bounds for the sum of domination number of a graph and its total graph and characterize the extremal graphs.

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

Strong vb-dominating and vb-independent sets of a graph

2019 ◽  
Vol 12 (01) ◽  
pp. 2050002 ◽  
Author(s):  
Sayinath Udupa ◽  
R. S. Bhat
Keyword(s):  
Lower Bounds ◽  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Independent Set ◽  
Independent Sets ◽  
Upper And Lower Bounds ◽  
Number Formula

Let [Formula: see text] be a graph. A vertex [Formula: see text] strongly (weakly) b-dominates block [Formula: see text] if [Formula: see text] ([Formula: see text]) for every vertex [Formula: see text] in the block [Formula: see text]. A set [Formula: see text] is said to be strong (weak) vb-dominating set (SVBD-set) (WVBD-set) if every block in [Formula: see text] is strongly (weakly) b-dominated by some vertex in [Formula: see text]. The strong (weak) vb-domination number [Formula: see text] ([Formula: see text]) is the order of a minimum SVBD (WVBD) set of [Formula: see text]. A set [Formula: see text] is said to be strong (weak) vertex block independent set (SVBI-set (WVBI-set)) if [Formula: see text] is a vertex block independent set and for every vertex [Formula: see text] and every block [Formula: see text] incident on [Formula: see text], there exists a vertex [Formula: see text] in the block [Formula: see text] such that [Formula: see text] ([Formula: see text]). The strong (weak) vb-independence number [Formula: see text] ([Formula: see text]) is the cardinality of a maximum strong (weak) vertex block independent set (SVBI-set) (WVBI-set) of [Formula: see text]. In this paper, we investigate some relationships between these four parameters. Several upper and lower bounds are established. In addition, we characterize the graphs attaining some of the bounds.

scholarly journals The Roman domination number of some special classes of graphs - convex polytopes

2021 ◽  
pp. 19-19
Author(s):  
Aleksandar Kartelj ◽  
Milana Grbic ◽  
Dragan Matic ◽  
Vladimir Filipovic
Keyword(s):  
Lower Bounds ◽  
Planar Graphs ◽  
Convex Polytopes ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Special Classes

In this paper we study the Roman domination number of some classes of planar graphs - convex polytopes: An, Rn and Tn. We establish the exact values of Roman domination number for: An, R3k, R3k+1, T8k, T8k+2, T8k+3, T8k+5 and T8k+6. For R3k+2, T8k+1, T8k+4 and T8k-1 we propose new upper and lower bounds, proving that the gap between the bounds is 1 for all cases except for the case of T8k+4, where the gap is 2.

Sign in / Sign up

Export Citation Format

Share Document