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.

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.

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 $k=1$.

Weak and strong domination in thorn graphs

2019 ◽  
Vol 13 (04) ◽  
pp. 2050071
Author(s):  
Derya Doğan Durgun ◽  
Berna Lökçü
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Number Formula ◽  
Thorn Graphs ◽  
Domination Numbers

Let [Formula: see text] be a graph and [Formula: see text] A dominating set [Formula: see text] is a set of vertices such that each vertex of [Formula: see text] is either in [Formula: see text] or has at least one neighbor in [Formula: see text]. The minimum cardinality of such a set is called the domination number of [Formula: see text], [Formula: see text] [Formula: see text] strongly dominates [Formula: see text] and [Formula: see text] weakly dominates [Formula: see text] if (i) [Formula: see text] and (ii) [Formula: see text] A set [Formula: see text] is a strong-dominating set, shortly sd-set, (weak-dominating set, shortly wd-set) of [Formula: see text] if every vertex in [Formula: see text] is strongly (weakly) dominated by at least one vertex in [Formula: see text]. The strong (weak) domination number [Formula: see text] of [Formula: see text] is the minimum cardinality of an sd-set (wd-set). In this paper, we present weak and strong domination numbers of thorn graphs.

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.

Disjunctive total domination in permutation graphs

2017 ◽  
Vol 09 (01) ◽  
pp. 1750009 ◽  
Author(s):  
Eunjeong Yi
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Permutation Graph ◽  
Permutation Graphs ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Vertex Set ◽  
Isolated Vertex ◽  
Number Formula

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. If [Formula: see text] has no isolated vertex, then a disjunctive total dominating set (DTD-set) of [Formula: see text] is a vertex set [Formula: see text] such that every vertex in [Formula: see text] is adjacent to a vertex of [Formula: see text] or has at least two vertices in [Formula: see text] at distance two from it, and the disjunctive total domination number [Formula: see text] of [Formula: see text] is the minimum cardinality overall DTD-sets of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be two disjoint copies of a graph [Formula: see text], and let [Formula: see text] be a bijection. Then, a permutation graph [Formula: see text] has the vertex set [Formula: see text] and the edge set [Formula: see text]. For any connected graph [Formula: see text] of order at least three, we prove the sharp bounds [Formula: see text]; we give an example showing that [Formula: see text] can be arbitrarily large. We characterize permutation graphs for which [Formula: see text] holds. Further, we show that [Formula: see text] when [Formula: see text] is a cycle, a path, and a complete [Formula: see text]-partite graph, respectively.

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.

scholarly journals Restrained Edge Domination Number of Some Path Related Graphs

2021 ◽  
Vol 13 (1) ◽  
pp. 145-151
Author(s):  
S. K. Vaidya ◽  
P. D. Ajani
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Restrained Domination ◽  
Edge Dominating Set ◽  
Middle Graph

For a graph G = (V,E), a set  S ⊆ V(S ⊆ E) is a restrained dominating (restrained edge dominating) set if every vertex (edge) not in S is adjacent (incident) to a vertex (edge) in S and to a vertex (edge) in V - S(E-S). The minimum cardinality of a restrained dominating (restrained edge dominating) set of G is called restrained domination (restrained edge domination) number of G, denoted by  γr (G) (γre(G). The restrained edge domination number of some standard graphs are already investigated while in this paper the restrained edge domination number like degree splitting, switching,  square and middle graph obtained from path.

scholarly journals On Two Open Problems on Double Vertex-Edge Domination in Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1010
Author(s):  
Fang Miao ◽  
Wenjie Fan ◽  
Mustapha Chellali ◽  
Rana Khoeilar ◽  
Seyed Mahmoud Sheikholeslami ◽  
...  
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Open Problems ◽  
Free Graph ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Edge Dominating Set ◽  
Domination In Graphs

A vertex v of a graph G = ( V , E ) , ve-dominates every edge incident to v, as well as every edge adjacent to these incident edges. A set S ⊆ V is a double vertex-edge dominating set if every edge of E is ve-dominated by at least two vertices of S. The double vertex-edge domination number γ d v e ( G ) is the minimum cardinality of a double vertex-edge dominating set in G. A subset S ⊆ V is a total dominating set (respectively, a 2-dominating set) if every vertex in V has a neighbor in S (respectively, every vertex in V - S has at least two neighbors in S). The total domination number γ t ( G ) is the minimum cardinality of a total dominating set of G, and the 2-domination number γ 2 ( G ) is the minimum cardinality of a 2-dominating set of G . Krishnakumari et al. (2017) showed that for every triangle-free graph G , γ d v e ( G ) ≤ γ 2 ( G ) , and in addition, if G has no isolated vertices, then γ d v e ( G ) ≤ γ t ( G ) . Moreover, they posed the problem of characterizing those graphs attaining the equality in the previous bounds. In this paper, we characterize all trees T with γ d v e ( T ) = γ t ( T ) or γ d v e ( T ) = γ 2 ( T ) .

Bounds on total edge domination number of a tree

2020 ◽  
pp. 2150011
Author(s):  
B. Senthilkumar ◽  
H. Naresh Kumar ◽  
Y. B. Venkatakrishnan
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Edge Dominating Set ◽  
Vertex Set

For a 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 the total edge dominating set if every edge in [Formula: see text] is adjacent to at least one edge in [Formula: see text]. The minimum cardinality of a total edge dominated set, denoted by [Formula: see text], is called the total edge domination number of a graph [Formula: see text]. We prove that for every tree [Formula: see text] of diameter at least two with [Formula: see text] leaves and [Formula: see text] support vertices we have [Formula: see text], and we characterize the trees attaining each of the bounds.

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.

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