scholarly journals Game $k$-Domination Number of Graphs

2021 ◽  
Vol 52 ◽  
Author(s):  
Rana Khoeilar ◽  
Mustapha Chellali ◽  
Hossein Karami ◽  
Seyed Mahmoud Sheikholeslami
Keyword(s):  
Positive Integer ◽  
Directed Graph ◽  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Dominating Sets ◽  
Minimum Cardinality

For a positive integer $k$, a subset $D$ of vertices in a digraph $\overrightarrow{G}$ is a $k$-dominating set if every vertex not in $D$ has at least $k$ direct predecessors in $D.$ The $k$-domination number is the minimum cardinality among all $k$-dominating sets of $\overrightarrow{G}$. The game $k$-domination number of a simple and undirected graph is defined by the following game. Two players, $\mathcal{A}$ and $\mathcal{D}$, orient the edges of the graph alternately until all edges are oriented. Player $\mathcal{D}$ starts the game, and his goal is to decrease the $k$-domination number of the resulting digraph, while $\mathcal{A}$ is trying to increase it. The game $k$-domination number of the graph $G$ is the $k$-domination number of the directed graph resulting from this game. This is well defined if we suppose that both players follow their optimal strateries. We are mainly interested in the study of the game $2$-domination number, where some upper bounds will be presented. We also establish a Nordhaus-Gaddum bound for the game $2$-domination number of a graph and its complement.

Pitchfork domination in graphs

2020 ◽  
Vol 12 (02) ◽  
pp. 2050025
Author(s):  
Manal N. Al-Harere ◽  
Mohammed A. Abdlhusein
Keyword(s):  
Maximum Degree ◽  
Undirected Graph ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
New Model ◽  
Domination In Graphs

In this paper, a new model of domination in graphs called the pitchfork domination is introduced. Let [Formula: see text] be a finite, simple and undirected graph without isolated vertices, a subset [Formula: see text] of [Formula: see text] is a pitchfork dominating set if every vertex [Formula: see text] dominates at least [Formula: see text] and at most [Formula: see text] vertices of [Formula: see text], where [Formula: see text] and [Formula: see text] are non-negative integers. The domination number of [Formula: see text], denotes [Formula: see text] is a minimum cardinality over all pitchfork dominating sets in [Formula: see text]. In this work, pitchfork domination when [Formula: see text] and [Formula: see text] is studied. Some bounds on [Formula: see text] related to the order, size, minimum degree, maximum degree of a graph and some properties are given. Pitchfork domination is determined for some known and new modified graphs. Finally, a question has been answered and discussed that; does every finite, simple and undirected graph [Formula: see text] without isolated vertices have a pitchfork domination or not?

The minimum eccentric distance sum of trees with given distance k-domination number

2020 ◽  
Vol 12 (04) ◽  
pp. 2050052 ◽  
Author(s):  
Lidan Pei ◽  
Xiangfeng Pan
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Simple Connected Graph ◽  
Eccentric Distance ◽  
Distance Formula

Let [Formula: see text] be a positive integer and [Formula: see text] be a simple connected graph. The eccentric distance sum of [Formula: see text] is defined as [Formula: see text], where [Formula: see text] is the maximum distance from [Formula: see text] to any other vertex and [Formula: see text] is the sum of all distances from [Formula: see text]. A set [Formula: see text] is a distance [Formula: see text]-dominating set of [Formula: see text] if for every vertex [Formula: see text], [Formula: see text] for some vertex [Formula: see text]. The minimum cardinality among all distance [Formula: see text]-dominating sets of [Formula: see text] is called the distance [Formula: see text]-domination number [Formula: see text] of [Formula: see text]. In this paper, the trees among all [Formula: see text]-vertex trees with distance [Formula: see text]-domination number [Formula: see text] having the minimal eccentric distance sum are determined.

scholarly journals The edge-to-edge geodetic domination number of a graph

2021 ◽  
Vol 40 (3) ◽  
pp. 635-658
Author(s):  
J. John ◽  
V. Sujin Flower
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Upper Bounds ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
General Properties ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Positive Integers

Let G = (V, E) be a connected graph with at least three vertices. A set S ⊆ E(G) is called an edge-to-edge geodetic dominating set of G if S is both an edge-to-edge geodetic set of G and an edge dominating set of G. The edge-to-edge geodetic domination number γgee(G) of G is the minimum cardinality of its edge-to-edge geodetic dominating sets. Some general properties satisfied by this concept are studied. Connected graphs of size m with edge-to-edge geodetic domination number 2 or m or m − 1 are characterized. We proved that if G is a connected graph of size m ≥ 4 and Ḡ is also connected, then 4 ≤ γgee(G) + γgee(Ḡ) ≤ 2m − 2. Moreover we characterized graphs for which the lower and the upper bounds are sharp. It is shown that, for every pair of positive integers a, b with 2 ≤ a ≤ b, there exists a connected graph G with gee(G) = a and γgee(G) = b. Also it is shown that, for every pair of positive integers a and b with 2 < a ≤ b, there exists a connected graph G with γe(G) = a and γgee(G) = b, where γe(G) is the edge domination number of G and gee(G) is the edge-to-edge geodetic number of G.

scholarly journals Stable Locating-Dominating Sets in Graphs

2021 ◽  
Vol 14 (3) ◽  
pp. 638-649
Author(s):  
Eman Camad Ahmad ◽  
Gina Alquiza Malacas ◽  
Sergio Jr. Rosales Canoy
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality

A set S ⊆ V(G) of a (simple) undirected graph G is a locating-dominating set of G if for each v ∈ V(G) \ S, there exists w ∈ S such tha vw ∈ E(G) and NG(x) ∩ S= NG(y)∩S for any distinct vertices x and y in V(G) \ S. S is a stable locating-dominating set of G if it is a locating-dominating set of G and S \ {v} is a locating-dominating set of G for each v ∈ S. The minimum cardinality of a stable locating-dominating set of G, denoted by γsl(G), is called the stable locating-domination number of G. In this paper, we investigate this concept and the corresponding parameter for some graphs. Further, we introduce other related concepts and use them to characterize the stable locating-dominating sets in some graphs.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{&#x0394;}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

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.

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 On the Total Outer k-Independent Domination Number of Graphs

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 194 ◽  
Author(s):  
Abel Cabrera-Martínez ◽  
Juan Carlos Hernández-Gómez ◽  
Ernesto Parra-Inza ◽  
José María Sigarreta Almira
Keyword(s):  
Maximum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Np Hard Problem ◽  
Independent Domination ◽  
Independent Dominating Set ◽  
Computational Properties

A set of vertices of a graph G is a total dominating set if every vertex of G is adjacent to at least one vertex in such a set. We say that a total dominating set D is a total outer k-independent dominating set of G if the maximum degree of the subgraph induced by the vertices that are not in D is less or equal to k − 1 . The minimum cardinality among all total outer k-independent dominating sets is the total outer k-independent domination number of G. In this article, we introduce this parameter and begin with the study of its combinatorial and computational properties. For instance, we give several closed relationships between this novel parameter and other ones related to domination and independence in graphs. In addition, we give several Nordhaus–Gaddum type results. Finally, we prove that computing the total outer k-independent domination number of a graph G is an NP-hard problem.

Complementary equitably totally disconnected equitable domination in graphs

2020 ◽  
pp. 2150043
Author(s):  
P. Nataraj ◽  
R. Sundareswaran ◽  
V. Swaminathan
Keyword(s):  
Undirected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Domination In Graphs ◽  
Totally Disconnected

In a simple, finite and undirected graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is said to be a degree equitable dominating set if for every [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], where [Formula: see text] denotes the degree of [Formula: see text] in [Formula: see text]. The minimum cardinality of such a dominating set is denoted by [Formula: see text] and is called the equitable domination number of [Formula: see text]. In this paper, we introduce Complementary Equitably Totally Disconnected Equitable domination in graphs and obtain some interesting results. Also, we discuss some bounds of this new domination parameter.

On the minimum vertex covering transversal dominating sets in graphs and their classification

2017 ◽  
Vol 09 (05) ◽  
pp. 1750069 ◽  
Author(s):  
R. Vasanthi ◽  
K. Subramanian
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Covering ◽  
Number Formula ◽  
Covering Set ◽  
Simple Connected Graph

Let [Formula: see text] be a simple and connected graph. A dominating set [Formula: see text] is said to be a vertex covering transversal dominating set if it intersects every minimum vertex covering set of [Formula: see text]. The vertex covering transversal domination number [Formula: see text] is the minimum cardinality among all vertex covering transversal dominating sets of [Formula: see text]. A vertex covering transversal dominating set of minimum cardinality [Formula: see text] is called a minimum vertex covering transversal dominating set or simply a [Formula: see text]-set. In this paper, we prove some general theorems on the vertex covering transversal domination number of a simple connected graph. We also provide some results about [Formula: see text]-sets and try to classify those sets based on their intersection with the minimum vertex covering sets.

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