scholarly journals A lower bound on the global powerful alliance number in trees

2021 ◽  
Author(s):  
Saliha Ouatiki ◽  
Mohamed Bouzefrane
Keyword(s):  
Lower Bound ◽  
Dominating Set ◽  
Minimum Cardinality ◽  
Closed Neighborhood

For a graph $G=(V,E)$, a set $D\subseteq V$ is a dominating set if every vertex in $V-D$ is either in $D$ or has a neighbor in $D$. A dominating set $D$ is a global offensive alliance (resp. a global defensive alliance) if for each vertex $v$ in $V-D$ (resp. $v$ in $D$) at least half the vertices from the closed neighborhood of $v$ are in $D$. A global powerful alliance is both global defensive and global offensive. The global powerful alliance number $\gamma_{pa}(G)$ is the minimum cardinality of a global powerful alliance of $G$. We show that if $T$ is a tree of order $n$ with $l$ leaves and $s$ support vertices, then $\gamma_{pa}(T)\geq\frac{3n-2l-s+2}{5}$. Moreover, we provide a constructive characterization of all extremal trees attaining this bound.

scholarly journals Graphs with Constant Sum of Domination and Inverse Domination Numbers

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
T. Asir
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Complete Characterization ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Vertex Set ◽  
Domination Numbers

A subset D of the vertex set of a graph G, is a dominating set if every vertex in V−D is adjacent to at least one vertex in D. The domination number γ(G) is the minimum cardinality of a dominating set of G. A subset of V−D, which is also a dominating set of G is called an inverse dominating set of G with respect to D. The inverse domination number γ′(G) is the minimum cardinality of the inverse dominating sets. Domke et al. (2004) characterized connected graphs G with γ(G)+γ′(G)=n, where n is the number of vertices in G. It is the purpose of this paper to give a complete characterization of graphs G with minimum degree at least two and γ(G)+γ′(G)=n−1.

scholarly journals Almost well-dominated bipartite graphs with minimum degree at least two

2020 ◽  
Author(s):  
Hadi Alizadeh ◽  
Didem Gözüpek
Keyword(s):  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
Upper Domination Number ◽  
The Difference ◽  
Two Parameters

A dominating set in a graph $G=(V,E)$ is a set $S$ such that every vertex of $G$ is either in $S$ or adjacent to a vertex in $S$. While the minimum cardinality of a dominating set in $G$ is called the domination number of $G$ denoted by $\gamma(G)$, the maximum cardinality of a minimal dominating set in $G$ is called the upper domination number of $G$ denoted by $\Gamma(G)$. We call the difference between these two parameters the \textit{domination gap} of $G$ and denote it by $\mu_d(G) = \Gamma(G) - \gamma(G)$. While a graph $G$ with $\mu_d(G)=0$ is said to be a \textit{well-dominated} graph, we call a graph $G$ with $\mu_d(G)=1$ an \textit{almost well-dominated} graph. In this work, we first establish an upper bound for the cardinality of bipartite graphs with $\mu_d(G)=k$, where $k\geq1$, and minimum degree at least two. We then provide a complete structural characterization of almost well-dominated bipartite graphs with minimum degree at least two. While the results by Finbow et al.~\cite{domination} imply that a 4-cycle is the only well-dominated bipartite graph with minimum degree at least two, we prove in this paper that there exist precisely 31 almost well-dominated bipartite graphs with minimum degree at least two.

Super domination in trees

2021 ◽  
pp. 2150137
Author(s):  
B. Senthilkumar ◽  
Y. B. Venkatakrishnan ◽  
H. Naresh Kumar
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality

Let [Formula: see text] be a simple graph. A set [Formula: see text] is called a super dominating set if for every vertex [Formula: see text], there exist [Formula: see text] such that [Formula: see text]. The minimum cardinality of a super dominating set of [Formula: see text], denoted by [Formula: see text], is called the super domination number of graph [Formula: see text]. Characterization of trees with [Formula: see text] is presented.

scholarly journals Trees with unique minimum glolal offensive alliance

2020 ◽  
Author(s):  
Mohamed Bouzefrane ◽  
Isma Bouchemakh ◽  
Mohamed Zamime ◽  
Noureddine Ikhlef-Eschouf
Keyword(s):  
Dominating Set ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Unique Minimum

Let G= (V,E) be a simple graph. A non-empty set D⊆V is called a global offensive alliance if D is a dominating set and for every vertex v in V-D, |N_{G}[v]∩D|≥|N_{G}[v]-D|. The global offensive alliance number is the minimum cardinality of a global offensive alliance in G. In this paper, we give a constructive characterization of trees having a unique minimum global offensive alliance.

2-domination dot-stable and dot-critical graphs

2019 ◽  
pp. 2150010
Author(s):  
Karima Attalah ◽  
Mustapha Chellali
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
The Other ◽  
Minimum Cardinality ◽  
Critical Graphs ◽  
Other Hand

A set [Formula: see text] of vertices in a graph [Formula: see text] is a [Formula: see text]-dominating set of [Formula: see text] if every vertex of [Formula: see text] is adjacent to at least two vertices in [Formula: see text] The [Formula: see text]-domination number is the minimum cardinality of a [Formula: see text]-dominating set of [Formula: see text]. A graph is [Formula: see text]-domination dot-stable if the contraction of any arbitrary edge has no effect on the [Formula: see text]-domination number. On the other hand, a graph is [Formula: see text]-domination dot-critical if the contraction of any arbitrary edge decreases the [Formula: see text]-domination number. We present some properties of these graphs and we provide a characterization of all trees that are [Formula: see text]-domination dot-stable or [Formula: see text]-domination dot-critical.

Characterization of some classes of graphs with equal domination number and isolate domination number

2020 ◽  
Vol 12 (05) ◽  
pp. 2050065
Author(s):  
Davood Bakhshesh
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Interval Graph ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Proper Interval Graph ◽  
Domination In Graphs

Let [Formula: see text] be a simple and undirected graph with vertex set [Formula: see text]. A set [Formula: see text] is called a dominating set of [Formula: see text], if every vertex in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text], denoted by [Formula: see text]. A dominating set [Formula: see text] of [Formula: see text] is called isolate dominating, if the induced subgraph [Formula: see text] of [Formula: see text] contains at least one isolated vertex. The minimum cardinality of an isolate dominating set of [Formula: see text] is called the isolate domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we show that for every proper interval graph [Formula: see text], [Formula: see text]. Moreover, we provide a constructive characterization for trees with equal domination number and isolate domination number. These solve part of an open problem posed by Hamid and Balamurugan [Isolate domination in graphs, Arab J. Math. Sci. 22(2) (2016) 232–241].

A lower bound on the total vertex-edge domination number of a tree

2020 ◽  
pp. 2050091
Author(s):  
B. Senthilkumar ◽  
H. Naresh Kumar ◽  
Y. B. Venkatakrishnan
Keyword(s):  
Lower Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Edge Incident ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Edge Dominating Set

A vertex [Formula: see text] of a graph [Formula: see text] is said to vertex-edge dominate every edge incident to [Formula: see text], as well as every edge adjacent to these incident edges. A subset [Formula: see text] is a vertex-edge dominating set (ve-dominating set) if every edge of [Formula: see text] is vertex-edge dominated by at least one vertex of [Formula: see text]. A vertex-edge dominating set is said to be total if its induced subgraph has no isolated vertices. The minimum cardinality of a total vertex-edge dominating set of [Formula: see text], denoted by [Formula: see text], is called the total vertex-edge domination number of [Formula: see text]. In this paper, we prove that for every nontrivial tree of order [Formula: see text], with [Formula: see text] leaves and [Formula: see text] support vertices we have [Formula: see text], and we characterize extremal trees attaining the lower bound.

Double vertex-edge domination

2017 ◽  
Vol 09 (04) ◽  
pp. 1750045 ◽  
Author(s):  
Balakrishna Krishnakumari ◽  
Mustapha Chellali ◽  
Yanamandram B. Venkatakrishnan
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Dominating Set ◽  
Domination Number ◽  
Edge Incident ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Number Formula ◽  
Number Of Leaves ◽  
Np Complete

A vertex [Formula: see text] of a graph [Formula: see text] is said to [Formula: see text]-dominate every edge incident to [Formula: see text], as well as every edge adjacent to these incident edges. A set [Formula: see text] is a vertex-edge dominating set (double vertex-edge dominating set, respectively) if every edge of [Formula: see text] is [Formula: see text]-dominated by at least one vertex (at least two vertices) of [Formula: see text] The minimum cardinality of a vertex-edge dominating set (double vertex-edge dominating set, respectively) of [Formula: see text] is the vertex-edge domination number [Formula: see text] (the double vertex-edge domination number [Formula: see text], respectively). In this paper, we initiate the study of double vertex-edge domination. We first show that determining the number [Formula: see text] for bipartite graphs is NP-complete. We also prove that for every nontrivial connected graphs [Formula: see text] [Formula: see text] and we characterize the trees [Formula: see text] with [Formula: see text] or [Formula: see text] Finally, we provide two lower bounds on the double ve-domination number of trees and unicycle graphs in terms of the order [Formula: see text] the number of leaves and support vertices, and we characterize the trees attaining the lower bound.

scholarly journals Secure Resolving Sets in a Graph

Symmetry ◽  
2018 ◽  
Vol 10 (10) ◽  
pp. 439
Author(s):  
Hemalathaa Subramanian ◽  
Subramanian Arasappan
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Resolving Set ◽  
Minimum Cardinality

Let G = (V, E) be a simple, finite, and connected graph. A subset S = {u1, u2, …, uk} of V(G) is called a resolving set (locating set) if for any x ∈ V(G), the code of x with respect to S that is denoted by CS (x), which is defined as CS (x) = (d(u1, x), d(u2, x), .., d(uk, x)), is different for different x. The minimum cardinality of a resolving set is called the dimension of G and is denoted by dim(G). A security concept was introduced in domination. A subset D of V(G) is called a dominating set of G if for any v in V – D, there exists u in D such that u and v are adjacent. A dominating set D is secure if for any u in V – D, there exists v in D such that (D – {v}) ∪ {u} is a dominating set. A resolving set R is secure if for any s ∈ V – R, there exists r ∈ R such that (R – {r}) ∪ {s} is a resolving set. The secure resolving domination number is defined, and its value is found for several classes of graphs. The characterization of graphs with specific secure resolving domination number is also done.

On defensive alliance in zero-divisor graphs

2020 ◽  
pp. 2150155
Author(s):  
Najat Muthana ◽  
Abdellah Mamouni
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Dominating Set ◽  
Complete Characterization ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Finite Commutative Ring ◽  
Multiple Edges

Let [Formula: see text] be a finite undirected graph without loops or multiple edges. A non-empty set of vertices [Formula: see text] is called defensive alliance if every vertex in [Formula: see text] have at least one more neighbors inside of [Formula: see text] than it has outside of [Formula: see text]. A non-empty set of vertices [Formula: see text] is called a dominating set if every vertex not in [Formula: see text] is adjacent to at least one member of [Formula: see text]. A defensive alliance dominating set is called global. The global defensive alliance number [Formula: see text] is defined as the minimum cardinality among all global defensive alliances. In this paper, we initiate the study of the global defensive alliance number of zero-divisor graphs [Formula: see text] with [Formula: see text] is a finite commutative ring. Hence, we calculate [Formula: see text] for some usual kind of rings. We finish by a complete characterization of rings with [Formula: see text].

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