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 On the upper geodetic global domination number of a graph

2020 ◽  
Vol 39 (6) ◽  
pp. 1627-1647
Author(s):  
X. Lenin Xaviour ◽  
S. Robinson Chellathurai
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Proper Subset ◽  
Maximum Cardinality ◽  
General Properties ◽  
Geodetic Set ◽  
Positive Integers ◽  
Global Domination Number

A set S of vertices in a connected graph G = (V, E) is called a geodetic set if every vertex not in S lies on a shortest path between two vertices from S. A set D of vertices in G is called a dominating set of G if every vertex not in D has at least one neighbor in D. A set D is called a global dominating set in G if S is a dominating set of both G and Ḡ. A set S is called a geodetic global dominating set of G if S is both geodetic and global dominating set of G. A geodetic global dominating set S in G is called a minimal geodetic global dominating set if no proper subset of S is itself a geodetic global dominating set in G. The maximum cardinality of a minimal geodetic global dominating set in G is the upper geodetic global domination number Ῡg+(G) of G. In this paper, the upper geodetic global domination number of certain connected graphs are determined and some of the general properties are studied. It is proved that for all positive integers a, b, p where 3 ≤ a ≤ b < p, there exists a connected graph G such that Ῡg(G) = a, Ῡg+(G) = b and |V (G)| = p.

scholarly journals Convex dominating-geodetic partitions in graphs

Filomat ◽  
2016 ◽  
Vol 30 (11) ◽  
pp. 3075-3082
Author(s):  
Yero González ◽  
Magdalena Lemńska
Keyword(s):  
Present Article ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Minimum Cardinality ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Convex Domination Number

The distance d(u,v) between two vertices u and v in a connected graph G is the length of a shortest u-v path in G. A u-v path of length d(u,v) is called u-v geodesic. A set X is convex in G if vertices from all a -b geodesics belong to X for every two vertices a,b?X. A set of vertices D is dominating in G if every vertex of V-D has at least one neighbor in D. The convex domination number con(G) of a graph G equals the minimum cardinality of a convex dominating set in G. A set of vertices S of a graph G is a geodetic set of G if every vertex v ? S lies on a x-y geodesic between two vertices x,y of S. The minimum cardinality of a geodetic set of G is the geodetic number of G and it is denoted by g(G). Let D,S be a convex dominating set and a geodetic set in G, respectively. The two sets D and S form a convex dominating-geodetic partition of G if |D| + |S| = |V(G)|. Moreover, a convex dominating-geodetic partition of G is called optimal if D is a ?con(G)-set and S is a g(G)-set. In the present article we study the (optimal) convex dominating-geodetic partitions of graphs.

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 $k=1$.

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.

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 Hop Dominating Sets in Graphs Under Binary Operations

2019 ◽  
Vol 12 (4) ◽  
pp. 1455-1463
Author(s):  
Sergio Canoy, Jr ◽  
Reynaldo Villarobe Mollejon ◽  
John Gabriel E. Canoy
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Binary Operations ◽  
Simple Connected Graph

Let G be a (simple) connected graph with vertex and edge sets V (G) and E(G),respectively. A set S ⊆ V (G) is a hop dominating set of G if for each v ∈ V (G) \ S, there exists w ∈ S such that dG(v, w) = 2. The minimum cardinality of a hop dominating set of G, denoted by γh(G), is called the hop domination number of G. In this paper we revisit the concept of hop domination, relate it with other domination concepts, and investigate it in graphs resulting from some binary operations.

The forcing near geodetic number of a graph

2020 ◽  
Vol 12 (02) ◽  
pp. 2050029
Author(s):  
R. Lenin
Keyword(s):  
Connected Graph ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Unique Minimum ◽  
Geodetic Number ◽  
Forcing Number ◽  
Geodetic Set ◽  
Number Formula ◽  
Positive Integers

A set [Formula: see text] is a near geodetic set if for every [Formula: see text] in [Formula: see text] there exist some [Formula: see text] in [Formula: see text] with [Formula: see text] The near geodetic number [Formula: see text] is the minimum cardinality of a near geodetic set in [Formula: see text] A subset [Formula: see text] of a minimum near geodetic set [Formula: see text] is called the forcing subset of [Formula: see text] if [Formula: see text] is the unique minimum near geodetic set containing [Formula: see text]. The forcing number [Formula: see text] of [Formula: see text] in [Formula: see text] is the minimum cardinality of a forcing subset for [Formula: see text], while the forcing near geodetic number [Formula: see text] of [Formula: see text] is the smallest forcing number among all minimum near geodetic sets of [Formula: see text]. In this paper, we initiate the study of forcing near geodetic number of connected graphs. We characterize graphs with [Formula: see text]. Further, we compare the parameters geodetic number[Formula: see text] near geodetic number[Formula: see text] forcing near geodetic number and we proved that, for every positive integers [Formula: see text] and [Formula: see text] with [Formula: see text] and [Formula: see text], there exists a nontrivial connected graph [Formula: see text] with [Formula: see text] [Formula: see text] and [Formula: see text].

scholarly journals Cost Effective Domination in the Join, Corona and Composition of Graphs

2019 ◽  
Vol 12 (3) ◽  
pp. 978-998
Author(s):  
Ferdinand P. Jamil ◽  
Hearty Nuenay Maglanque
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Cost Effective ◽  
Minimal Cost ◽  
Dominating Sets ◽  
Maximum Cardinality ◽  
Minimum Cardinality ◽  
The Cost ◽  
Domination Numbers

Let $G$ be a connected graph. A cost effective dominating set in a graph $G$ is any set $S$ of vertices in $G$ satisfying the condition that each vertex in $S$ is adjacent to at least as many vertices outside $S$ as inside $S$ and every vertex outside $S$ is adjacent to at least one vertex in $S$. The minimum cardinality of a cost effective dominating set is the cost effective domination number of $G$. The maximum cardinality of a cost effective dominating set is the upper cost effective domination number of $G$. A cost effective dominating set is said to be minimal if it does not contain a proper subset which is itself a cost effective dominating in $G$. The maximum cardinality of a minimal cost effective dominating set in a graph $G$ is the minimal cost effective domination number of $G$.In this paper, we characterized the cost effective dominating sets in the join, corona and composition of graphs. As direct consequences, we the bounds or the exact cost effective domination numbers, minimal cost effective domination numbers and upper cost effective domination numbers of these graphs were obtained.

scholarly journals Claw-free graphs with equal 2-domination and domination numbers

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2795-2801 ◽  
Author(s):  
Adriana Hansberg ◽  
Bert Randerath ◽  
Lutz Volkmann
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Line Graphs ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Domination Numbers

For a graph G a subset D of the vertex set of G is a k-dominating set if every vertex not in D has at least k neighbors in D. The k-domination number k(G) is the minimum cardinality among the k-dominating sets of G. Note that the 1-domination number 1(G) is the usual domination number (G). Fink and Jacobson showed in 1985 that the inequality ?k(G)?(G)+k?2 is valid for every connected graph G. In this paper, we concentrate on the case k = 2, where k can be equal to ?, and we characterize all claw-free graphs and all line graphs G with ?(G) = ?2(G).

EXTREME STEINER GRAPHS

2012 ◽  
Vol 04 (02) ◽  
pp. 1250029 ◽  
Author(s):  
A. P. SANTHAKUMARAN
Keyword(s):  
Connected Graph ◽  
Steiner Tree ◽  
Minimum Order ◽  
Minimum Cardinality ◽  
Geodetic Number ◽  
Geodetic Set ◽  
Positive Integers

For a connected graph G of order p ≥ 2 and a set W ⊆ V(G), a tree T contained in G is a Steiner tree with respect to W if T is a tree of minimum order with W ⊆ V(T). The set S(W) consists of all vertices in G that lie on some Steiner tree with respect to W. The set W is a Steiner set for G if S(W) = V(G). The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. A geodetic set of G is a set S of vertices such that every vertex of G is contained in a geodesic joining some pair of vertices of S. The geodetic number g(G) of G is the minimum cardinality of its geodetic sets and any geodetic set of cardinality g(G) is a minimum geodetic set of G. A vertex v is an extreme vertex of a graph G if the subgraph induced by its neighbors is complete. The number of extreme vertices in G is its extreme order ex (G). A graph G is an extreme Steiner graph if s(G) = ex (G), and an extreme geodesic graph if g(G) = ex (G). Extreme Steiner graphs of order p with Steiner number p - 1 are characterized. It is shown that every pair a, b of integers with 0 ≤ a ≤ b is realizable as the extreme order and Steiner number, respectively, of some graph. For positive integers r, d and l ≥ 2 with r < d ≤ 2r, it is shown that there exists an extreme Steiner graph G of radius r, diameter d, and Steiner number l. For integers p, d and k with 2 ≤ d < p, 2 ≤ k < p and p - d - k + 2 ≥ 0, there exists an extreme Steiner graph G of order p, diameter d and Steiner number k. It is shown that for every pair a, b of integers with 3 ≤ a < b and b = a + 1, there exists an extreme Steiner graph G with s(G) = a and g(G) = b that is not an extreme geodesic graph. It is shown that for every pair a, b of integers with 3 ≤ a < b, there exists an extreme geodesic graph G with g(G) = a and s(G) = b that is not an extreme Steiner graph.

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