scholarly journals Bound of Distance Domination Number of Graph and Edge Comb Product Graph

2017 ◽  
Vol 855 ◽  
pp. 012014 ◽  
Author(s):  
A.W. Gembong ◽  
Slamin ◽  
Dafik ◽  
Ika Hesti Agustin
Keyword(s):  
Domination Number ◽  
Product Graph ◽  
Distance 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 $k=1$.

Trees with the same distance domination number

2020 ◽  
Vol 15 (2) ◽  
pp. 91-96
Author(s):  
Min-Jen Jou ◽  
Jenq-Jong Lin ◽  
Qian-Yu Lin
Keyword(s):  
Domination Number ◽  
Distance Domination

scholarly journals On the distance domination number of bipartite graphs

2020 ◽  
Vol 8 (2) ◽  
pp. 353
Author(s):  
Doost Ali Mojdeh ◽  
Seyed Reza Musawi ◽  
Esmaeil Nazari
Keyword(s):  
Domination Number ◽  
Bipartite Graphs ◽  
Distance Domination

scholarly journals Optimization Models for Efficient (t, r) Broadcast Domination in Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1028
Author(s):  
Poompol Buathong ◽  
Tipaluck Krityakierne
Keyword(s):  
Biological Networks ◽  
Domination Number ◽  
Real Life ◽  
Point Of View ◽  
Optimization Approach ◽  
Grid Graphs ◽  
Theoretical Approaches ◽  
Graphs And Networks ◽  
Domination In Graphs ◽  
Distance Domination

Known to be NP-complete, domination number problems in graphs and networks arise in many real-life applications, ranging from the design of wireless sensor networks and biological networks to social networks. Initially introduced by Blessing et al., the (t,r) broadcast domination number is a generalization of the distance domination number. While some theoretical approaches have been addressed for small values of t,r in the literature; in this work, we propose an approach from an optimization point of view. First, the (t,r) broadcast domination number is formulated and solved using linear programming. The efficient broadcast, whose wasted signals are minimized, is then found by a genetic algorithm modified for a binary encoding. The developed method is illustrated with several grid graphs: regular, slant, and king’s grid graphs. The obtained computational results show that the method is able to find the exact (t,r) broadcast domination number, and locate an efficient broadcasting configuration for larger values of t,r than what can be provided from a theoretical basis. The proposed optimization approach thus helps overcome the limitations of existing theoretical approaches in graph theory.

Exponential Domination Critical and Stability in Some Graphs

2019 ◽  
Vol 30 (05) ◽  
pp. 781-791 ◽  
Author(s):  
Aysun Aytaç ◽  
Betül Atay Atakul
Keyword(s):  
Shortest Path ◽  
Dominating Set ◽  
Domination Number ◽  
Number Formula ◽  
Distance Domination

An exponential dominating set of graph [Formula: see text] is a kind of distance domination subset [Formula: see text] such that [Formula: see text], [Formula: see text], where [Formula: see text] is the length of a shortest path in [Formula: see text] if such a path exists, and [Formula: see text] otherwise. The minimum exponential domination number, [Formula: see text] is the smallest cardinality of an exponential dominating set. The minimum exponential domination number, [Formula: see text] can be decreased or increased by removal of some vertices from [Formula: see text]. In this paper, we investigate of this phenomenon which is referred to critical and stability in graphs.

scholarly journals Dot product graphs and domination number

2020 ◽  
Vol 28 (1) ◽  
Author(s):  
Dina Saleh ◽  
Nefertiti Megahed
Keyword(s):  
Commutative Ring ◽  
Upper Bound ◽  
Undirected Graph ◽  
Multiplicative Group ◽  
Domination Number ◽  
Product Graph ◽  
Group Of Units ◽  
Product Graphs ◽  
Vertex Set ◽  
Dot Product

Abstract Let A be a commutative ring with 1≠0 and R=A×A. The unit dot product graph of R is defined to be the undirected graph UD(R) with the multiplicative group of units in R, denoted by U(R), as its vertex set. Two distinct vertices x and y are adjacent if and only if x·y=0∈A, where x·y denotes the normal dot product of x and y. In 2016, Abdulla studied this graph when $A=\mathbb {Z}_{n}$ A = ℤ n , $n \in \mathbb {N}$ n ∈ ℕ , n≥2. Inspired by this idea, we study this graph when A has a finite multiplicative group of units. We define the congruence unit dot product graph of R to be the undirected graph CUD(R) with the congruent classes of the relation $\thicksim $ ∽ defined on R as its vertices. Also, we study the domination number of the total dot product graph of the ring $R=\mathbb {Z}_{n}\times... \times \mathbb {Z}_{n}$ R = ℤ n × ... × ℤ n , k times and k<∞, where all elements of the ring are vertices and adjacency of two distinct vertices is the same as in UD(R). We find an upper bound of the domination number of this graph improving that found by Abdulla.

scholarly journals Distance Domination Number of Graphs Resulting from Edge Comb Product

2018 ◽  
Vol 1022 ◽  
pp. 012008
Author(s):  
Slamin ◽  
Dafik ◽  
Gembong Angger Waspodo
Keyword(s):  
Domination Number ◽  
Distance Domination
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