scholarly journals Properties of the Global Total k-Domination Number

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 480
Author(s):  
Frank A. Hernández Mira ◽  
Ernesto Parra Inza ◽  
José M. Sigarreta Almira ◽  
Nodari Vakhania
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Minimum Cardinality

A nonempty subset D⊂V of vertices of a graph G=(V,E) is a dominating set if every vertex of this graph is adjacent to at least one vertex from this set except the vertices which belong to this set itself. D⊆V is a total k-dominating set if there are at least k vertices in set D adjacent to every vertex v∈V, and it is a global total k-dominating set if D is a total k-dominating set of both G and G¯. The global total k-domination number of G, denoted by γktg(G), is the minimum cardinality of a global total k-dominating set of G, GTkD-set. Here we derive upper and lower bounds of γktg(G), and develop a method that generates a GTkD-set from a GT(k−1)D-set for the successively increasing values of k. Based on this method, we establish a relationship between γ(k−1)tg(G) and γktg(G), which, in turn, provides another upper bound on γktg(G).

Bounds on 2-point set domination number of a graph

2018 ◽  
Vol 10 (01) ◽  
pp. 1850012
Author(s):  
Purnima Gupta ◽  
Deepti Jain
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Extremal Graphs ◽  
Minimum Cardinality ◽  
Point Set

A set [Formula: see text] is a [Formula: see text]-point set dominating set (2-psd set) of a graph [Formula: see text] if for any subset [Formula: see text], there exists a nonempty subset [Formula: see text] containing at most two vertices such that the subgraph [Formula: see text] induced by [Formula: see text] is connected. The [Formula: see text]-point set domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a 2-psd set of [Formula: see text]. In this paper, we determine the lower bounds and an upper bound on [Formula: see text] of a graph. We also characterize extremal graphs for the lower bounds and identify some well-known classes of both separable and nonseparable graphs attaining the upper bound.

A note on improved upper bounds on the transversal number of hypergraphs

2019 ◽  
Vol 11 (01) ◽  
pp. 1950004
Author(s):  
Michael A. Henning ◽  
Nader Jafari Rad
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Size ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Isolated Vertex ◽  
Transversal Number

A subset [Formula: see text] of vertices in a hypergraph [Formula: see text] is a transversal if [Formula: see text] has a nonempty intersection with every edge of [Formula: see text]. The transversal number of [Formula: see text] is the minimum size of a transversal in [Formula: see text]. A subset [Formula: see text] of vertices in a graph [Formula: see text] with no isolated vertex, is a total dominating set if every vertex of [Formula: see text] is adjacent to a vertex of [Formula: see text]. The minimum cardinality of a total dominating set in [Formula: see text] is the total domination number of [Formula: see text]. In this paper, we obtain a new (improved) probabilistic upper bound for the transversal number of a hypergraph, and a new (improved) probabilistic upper bound for the total domination number of a graph.

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 $k=1$.

2-Point set domination in separable graphs

2021 ◽  
Author(s):  
Purnima Gupta ◽  
Deepti Jain
Keyword(s):  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Induced Subgraph ◽  
Minimum Cardinality ◽  
Point Set ◽  
Separable Graph

In a graph [Formula: see text], a set [Formula: see text] is a [Formula: see text]-point set dominating set (in short 2-psd set) of [Formula: see text] if for every subset [Formula: see text] there exists a nonempty subset [Formula: see text] containing at most two vertices such that the induced subgraph [Formula: see text] is connected in [Formula: see text]. The [Formula: see text]-point set domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of a 2-psd set of [Formula: see text]. The main focus of this paper is to find the value of [Formula: see text] for a separable graph and thereafter computing [Formula: see text] for some well-known classes of separable graphs. Further we classify the set of all 2-psd sets of a separable graph into six disjoint classes and study the existence of minimum 2-psd sets in each class.

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.

Total edge–vertex domination

2020 ◽  
Vol 54 ◽  
pp. 1 ◽  
Author(s):  
Abdulgani Sahin ◽  
Bünyamin Sahin
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Np Hard ◽  
Minimum Cardinality

An edge e ev-dominates a vertex v which is a vertex of e, as well as every vertex adjacent to v. A subset D ⊆ E is an edge-vertex dominating set (in simply, ev-dominating set) of G, if every vertex of a graph G is ev-dominated by at least one edge of D. The minimum cardinality of an ev-dominating set is named with ev-domination number and denoted by γev(G). A subset D ⊆ E is a total edge-vertex dominating set (in simply, total ev-dominating set) of G, if D is an ev-dominating set and every edge of D shares an endpoint with other edge of D. The total ev-domination number of a graph G is denoted with γevt(G) and it is equal to the minimum cardinality of a total ev-dominating set. In this paper, we initiate to study total edge-vertex domination. We first show that calculating the number γevt(G) for bipartite graphs is NP-hard. We also show the upper bound γevt(T) ≤ (n − l + 2s − 1)∕2 for the total ev-domination number of a tree T, where T has order n, l leaves and s support vertices and we characterize the trees achieving this upper bound. Finally, we obtain total ev-domination number of paths and cycles.

scholarly journals A New Upper Bound on the Total Domination Number of a Graph

10.37236/983 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set

A set $S$ of vertices in a graph $G$ is a total dominating set of $G$ if every vertex of $G$ is adjacent to some vertex in $S$. The minimum cardinality of a total dominating set of $G$ is the total domination number of $G$. Let $G$ be a connected graph of order $n$ with minimum degree at least two and with maximum degree at least three. We define a vertex as large if it has degree more than $2$ and we let ${\cal L}$ be the set of all large vertices of $G$. Let $P$ be any component of $G - {\cal L}$; it is a path. If $|P| \equiv 0 \, ( {\rm mod} \, 4)$ and either the two ends of $P$ are adjacent in $G$ to the same large vertex or the two ends of $P$ are adjacent to different, but adjacent, large vertices in $G$, we call $P$ a $0$-path. If $|P| \ge 5$ and $|P| \equiv 1 \, ( {\rm mod} \, 4)$ with the two ends of $P$ adjacent in $G$ to the same large vertex, we call $P$ a $1$-path. If $|P| \equiv 3 \, ( {\rm mod} \, 4)$, we call $P$ a $3$-path. For $i \in \{0,1,3\}$, we denote the number of $i$-paths in $G$ by $p_i$. We show that the total domination number of $G$ is at most $(n + p_0 + p_1 + p_3)/2$. This result generalizes a result shown in several manuscripts (see, for example, J. Graph Theory 46 (2004), 207–210) which states that if $G$ is a graph of order $n$ with minimum degree at least three, then the total domination of $G$ is at most $n/2$. It also generalizes a result by Lam and Wei stating that if $G$ is a graph of order $n$ with minimum degree at least two and with no degree-$2$ vertex adjacent to two other degree-$2$ vertices, then the total domination of $G$ is at most $n/2$.

The k-power bondage number of a graph

2016 ◽  
Vol 08 (04) ◽  
pp. 1650064
Author(s):  
Seethu Varghese ◽  
A. Vijayakumar
Keyword(s):  
Upper Bound ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Power Domination ◽  
Bondage Number ◽  
Number Formula

The [Formula: see text]-power domination number, [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of a [Formula: see text]-power dominating set of [Formula: see text]. In this paper, we initiate the study of the [Formula: see text]-power bondage number, [Formula: see text], of a graph [Formula: see text], i.e., the minimum cardinality among all sets [Formula: see text] for which [Formula: see text]. We obtain a sharp upper bound for [Formula: see text] in terms of the degree of [Formula: see text]. We prove that [Formula: see text] for any nonempty tree [Formula: see text] and also provide some conditions on [Formula: see text] for [Formula: see text].

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.

The minimum vertex–vertex dominating Laplacian energy of a graph

2021 ◽  
Author(s):  
N. V. Sayinath Udupa ◽  
R. S. Bhat
Keyword(s):  
Lower Bounds ◽  
Dominating Set ◽  
Domination Number ◽  
Upper And Lower Bounds ◽  
Underlying Graph ◽  
Laplacian Energy ◽  
Graph Energy ◽  
Number Formula

Let [Formula: see text] denote the set of all blocks of a graph [Formula: see text]. Two vertices are said to vv-dominate each other if they are vertices of the same block. A set [Formula: see text] is said to be vertex–vertex dominating set (vv-dominating set) if every vertex in [Formula: see text] is vv-dominated by some vertex in [Formula: see text]. The vv-domination number [Formula: see text] is the cardinality of the minimum vv-dominating set of [Formula: see text]. In this paper, we introduce new kind of graph energy, the minimum vv-dominating Laplacian energy of a graph denoting it as LE[Formula: see text]. It depends both on the underlying graph of [Formula: see text] and the particular minimum vv-dominating set of [Formula: see text]. Upper and lower bounds for LE[Formula: see text] are established and we also obtain the minimum vv-dominating Laplacian energy of some family of graphs.

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