Clique doubly connected domination in the join and lexicographic product of graphs

2020 ◽  
Vol 12 (05) ◽  
pp. 2050066
Author(s):  
Enrico L. Enriquez ◽  
Albert D. Ngujo
Keyword(s):  
Nonempty Subset ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Connected Dominating Set ◽  
Lexicographic Product ◽  
Connected Domination Number ◽  
Vertex Set ◽  
Product Of Graphs

Let [Formula: see text] be a connected simple graph. A set [Formula: see text] is a doubly connected dominating set if it is dominating and both [Formula: see text] and [Formula: see text] are connected. A nonempty subset [Formula: see text] of the vertex set [Formula: see text] is a clique in [Formula: see text] if the graph [Formula: see text] induced by [Formula: see text] is complete. A clique dominating set [Formula: see text] of [Formula: see text] is a clique doubly connected dominating set if [Formula: see text] is a doubly connected dominating set of [Formula: see text]. The clique doubly connected domination number of [Formula: see text], denoted by [Formula: see text], is the smallest cardinality of a clique doubly connected dominating set [Formula: see text] of [Formula: see text]. In this paper, we give the characterization of the clique doubly connected dominating set and the clique doubly connected domination number in the join (and lexicographic product) of two graphs.

Connected hop domination in graphs under some binary operations

2018 ◽  
Vol 11 (05) ◽  
pp. 1850075
Author(s):  
Yamilita M. Pabilona ◽  
Helen M. Rara
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Induced Subgraph ◽  
Binary Operations ◽  
Network Application ◽  
Domination In Graphs ◽  
Product Of Graphs

Let [Formula: see text] be a simple graph. A hop dominating set [Formula: see text] is called a connected hop dominating set of [Formula: see text] if the induced subgraph [Formula: see text] of [Formula: see text] is connected. The smallest cardinality of a connected hop dominating set of [Formula: see text], denoted by [Formula: see text], is called the connected hop domination number of [Formula: see text]. In this paper, we characterize the connected hop dominating sets in the join, corona and lexicographic product of graphs and determine the corresponding connected hop domination number of these graphs. The study of these concepts is motivated with a social network application.

scholarly journals Total Perfect Hop Domination in Graphs Under Some Binary Operations

2021 ◽  
Vol 14 (3) ◽  
pp. 803-815
Author(s):  
Raicah Cayongcat Rakim ◽  
Helen M Rara
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Binary Operations ◽  
Domination In Graphs ◽  
Product Of Graphs

Let G = (V (G), E(G)) be a simple graph. A set S ⊆ V (G) is a perfect hop dominating set of G if for every v ∈ V (G) \ S, there is exactly one vertex u ∈ S such that dG(u, v) = 2. The smallest cardinality of a perfect hop dominating set of G is called the perfect hop domination number of G, denoted by γph(G). A perfect hop dominating set S ⊆ V (G) is called a total perfect hop dominating set of G if for every v ∈ V (G), there is exactly one vertex u ∈ S such that dG(u, v) = 2. The total perfect hop domination number of G, denoted by γtph(G), is the smallest cardinality of a total perfect hop dominating set of G. Any total perfect hop dominating set of G of cardinality γtph(G) is referred to as a γtph-set of G. In this paper, we characterize the total perfect hop dominating sets in the join, corona and lexicographic product of graphs and determine their corresponding total perfect hop domination number.

Outer-paired domination in graphs

2020 ◽  
Vol 12 (06) ◽  
pp. 2050072
Author(s):  
A. Mahmoodi ◽  
L. Asgharsharghi
Keyword(s):  
Perfect Matching ◽  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Minimum Cardinality ◽  
Sharp Bounds ◽  
Paired Domination Number ◽  
Vertex Set ◽  
Paired Domination ◽  
Domination In Graphs

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text]. An outer-paired dominating set [Formula: see text] of a graph [Formula: see text] is a dominating set such that the subgraph induced by [Formula: see text] has a perfect matching. The outer-paired domination number of [Formula: see text], denoted by [Formula: see text], is the minimum cardinality of an outer-paired dominating set of [Formula: see text]. In this paper, we study the outer-paired domination number of graphs and present some sharp bounds concerning the invariant. Also, we characterize all the trees with [Formula: see text].

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.

A SIMPLE HEURISTIC FOR MINIMUM CONNECTED DOMINATING SET IN GRAPHS

2003 ◽  
Vol 14 (02) ◽  
pp. 323-333 ◽  
Author(s):  
PENG-JUN WAN ◽  
KHALED M. ALZOUBI ◽  
OPHIR FRIEDER
Keyword(s):  
Planar Graph ◽  
Complete Graph ◽  
Dominating Set ◽  
Independence Number ◽  
Domination Number ◽  
Connected Dominating Set ◽  
Minimum Connected Dominating Set ◽  
Simple Heuristic ◽  
Connected Domination Number ◽  
A Minor

Let α2(G), γ(G) and γc(G) be the 2-independence number, the domination number, and the connected domination number of a graph G respectively. Then α2(G) ≤ γ (G) ≤ γc(G). In this paper , we present a simple heuristic for Minimum Connected Dominating Set in graphs. When running on a graph G excluding Km (the complete graph of order m) as a minor, the heuristic produces a connected dominating set of cardinality at most 7α2(G) - 4 if m = 3, or at most [Formula: see text] if m ≥ 4. In particular, if running on a planar graph G, the heuristic outputs a connected dominating set of cardinality at most 15α2(G) - 5.

scholarly journals Minimum Total Dominating Energy of Some Standard Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 272-276
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Total Domination ◽  
Total Domination Number ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Ladder Graph ◽  
Eigen Values ◽  
Vertex Set

Let G be a simple graph with vertex set V(G) and edge set E(G). A set S of vertices in a graph 𝑮(𝑽,𝑬) is called a total dominating set if every vertex 𝒗 ∈ 𝑽 is adjacent to an element of S. The minimum cardinality of a total dominating set of G is called the total domination number of G which is denoted by 𝜸𝒕 (𝑮). The energy of the graph is defined as the sum of the absolute values of the eigen values of the adjacency matrix. In this paper, we computed minimum total dominating energy of a Friendship Graph, Ladder Graph and Helm graph. The Minimum total dominating energy for bistar graphand sun graph is also determined.

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 On the doubly connected domination number of a graph

2006 ◽  
Vol 4 (1) ◽  
pp. 34-45 ◽  
Author(s):  
Joanna Cyman ◽  
Magdalena Lemańska ◽  
Joanna Raczek
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Connected Dominating Set ◽  
Dominating Sets ◽  
Connected Dominating Sets ◽  
Connected Domination Number

AbstractFor a given connected graph G = (V, E), a set $$D \subseteq V(G)$$ is a doubly connected dominating set if it is dominating and both 〈D〉 and 〈V (G)-D〉 are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.

scholarly journals Liar’s domination in graphs under some operations

2017 ◽  
Vol 48 (1) ◽  
pp. 61-71 ◽  
Author(s):  
Sergio Jr. Rosales Canoy ◽  
Carlito Bancoyo Balandra
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Lexicographic Product ◽  
Domination In Graphs ◽  
Product Of Graphs

A set $S\subseteq V(G)$ is a liar's dominating set ($lds$) of graph $G$ if $|N_G[v]\cap S|\geq 2$ for every $v\in V(G)$ and $|(N_G[u]\cup N_G[v])\cap S|\geq 3$ for any two distinct vertices $u,v \in V(G)$. The liar's domination number of $G$, denoted by $\gamma_{LR}(G)$, is the smallest cardinality of a liar's dominating set of $G$. In this paper we study the concept of liar's domination in the join, corona, and lexicographic product of graphs.

scholarly journals Equitable Power Domination Number of Mycielskian of Certain Graphs

2018 ◽  
Vol 7 (4.10) ◽  
pp. 842 ◽  
Author(s):  
S. Banu Priya ◽  
A. Parthiban ◽  
N. Srinivasan
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Simple Graph ◽  
Adjacent Vertex ◽  
Minimum Cardinality ◽  
Power Domination ◽  
Vertex Set ◽  
The Difference

Let  be a simple graph with vertex set  and edge set . A set  is called a power dominating set (PDS), if every vertex   is observed by some vertices in  by using the following rules: (i) if a vertex  in  is in PDS, then it dominates itself and all the adjacent vertices of  and (ii) if an observed vertex  in   has  adjacent vertices and if   of these vertices are already observed, then the remaining one non-observed vertex is also observed by  in . A power dominating set    in   is said to be an equitable power dominating set (EPDS), if for every  there exists an adjacent vertex   such that the difference between the degree of  and degree of  is less than or equal to 1, i.e., . The minimum cardinality of an equitable power dominating set of  is called the equitable power domination number of  and denoted by . The Mycielskian of a graph  is the graph  with vertex set  where , and edge set  In this paper we investigate the equitable power domination number of Mycielskian of certain graphs. 

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