SECURE DOMINATING SETS IN THE LEXICOGRAPHIC PRODUCT OF GRAPHS

2019 ◽  
Vol 20 (1) ◽  
pp. 13-24
Author(s):  
Sergio R. Canoy ◽  
Seanne Abigail E. Canoy ◽  
Marlon F. Cruzate
Keyword(s):  
Lexicographic Product ◽  
Dominating Sets ◽  
Product Of Graphs

scholarly journals Forcing Subsets for γc-sets and γt-sets in the Lexicographic Product of Graphs

2019 ◽  
Vol 12 (4) ◽  
pp. 1779-1786
Author(s):  
Cris Laquibla Armada ◽  
Sergio, Jr. R. Canoy ◽  
Carmelito E. Go
Keyword(s):  
Total Domination ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Connected Dominating Sets ◽  
Product Of Graphs ◽  
Domination Numbers

In this paper, the connected dominating sets and total dominating sets in the lexicographic product of two graphs are characterized. Further, the connected domination, total domination, forcing connected domination and forcing total domination numbers of these graphs are determined.

scholarly journals Resolving Restrained Domination in Graphs

2021 ◽  
Vol 14 (3) ◽  
pp. 829-841
Author(s):  
Gerald Bacon Monsanto ◽  
Helen M. Rara
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Restrained Domination ◽  
Isolated Vertex ◽  
Domination In Graphs ◽  
Product Of Graphs ◽  
Restrained Domination Number

Let G be a connected graph. Brigham et al. [3] defined a resolving dominating setas a set S of vertices of a connected graph G that is both resolving and dominating. A set S ⊆ V (G) is a resolving restrained dominating set of G if S is a resolving dominating set of G and S = V (G) or hV (G) \ Si has no isolated vertex. In this paper, we characterize the resolving restrained dominating sets in the join, corona and lexicographic product of graphs and determine the resolving restrained domination number of these 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 Dierentiating-Dominating sets in graphs Under binary operations

2015 ◽  
Vol 46 (1) ◽  
pp. 51-60
Author(s):  
Sergio R.osales Canoy,Jr. ◽  
Gina A. Malacas
Keyword(s):  
Lexicographic Product ◽  
Dominating Sets ◽  
Binary Operations ◽  
Product Of Graphs ◽  
Domination Numbers

In this paper we characterize the differentiating-dominating sets in the join, corona, and lexicographic product of graphs. We also determine bounds or the exact differentiating-domination numbers of these graphs.

scholarly journals On Semitotal Domination in Graphs

2019 ◽  
Vol 12 (4) ◽  
pp. 1410-1425
Author(s):  
Imelda S. Aniversario ◽  
Sergio R. Canoy Jr. ◽  
Ferdinand P. Jamil
Keyword(s):  
Connected Graph ◽  
Dominating Set ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Semitotal Domination ◽  
Domination In Graphs ◽  
Product Of Graphs ◽  
Domination Numbers

A set $S$ of vertices of a connected graph $G$ is a semitotal dominating set if every vertex in $V(G)\setminus S$ is adjacent to a vertex in $S$, and every vertex in $S$ is of distance at most $2$ from another vertex in $S$. A semitotal dominating set $S$ in $G$ is a secure semitotal dominating set if for every $v\in V(G)\setminus S$, there is a vertex $x\in S$ such that $x$ is adjacent to $v$ and  that $\left(S\setminus\{x\}\right)\cup \{v\}$ is a semitotal dominating set in $G$. In this paper, we characterize the semitotal dominating sets and the secure semitotal dominating sets in the join, corona and lexicographic product of graphs and determine their corresponding semitotal domination and secure semitotal domination numbers.

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.

scholarly journals On Semitotal k-Fair and Independent k-Fair Domination in Graphs

2020 ◽  
Vol 13 (4) ◽  
pp. 779-793
Author(s):  
Marivir Ortega ◽  
Rowena Isla
Keyword(s):  
Positive Integer ◽  
Domination Number ◽  
Cartesian Product ◽  
Lexicographic Product ◽  
Dominating Sets ◽  
Sharp Bounds ◽  
Cartesian Product Of Graphs ◽  
Domination In Graphs ◽  
Product Of Graphs

In this paper, we introduce and investigate the concepts of semitotal k-fair domination and independent k-fair domination, where k is a positive integer. We also characterize the semitotal 1-fair dominating sets and independent k-fair dominating sets in the join, corona, lexicographic product, and Cartesian product of graphs and determine the exact value or sharp bounds of the corresponding semitotal 1-fair domination number and independent k-fair domination number.

scholarly journals Another H-super magic decompositions of the lexicographic product of graphs

2018 ◽  
Vol 2 (2) ◽  
pp. 72
Author(s):  
H Hendy ◽  
Kiki A. Sugeng ◽  
A.N.M Salman ◽  
Nisa Ayunda
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graph Decomposition ◽  
Graph Labeling ◽  
Sufficient Condition ◽  
Lexicographic Product ◽  
Simple Graphs ◽  
Product Of Graphs

<p>Let <span class="math"><em>H</em></span> and <span class="math"><em>G</em></span> be two simple graphs. The concept of an <span class="math"><em>H</em></span>-magic decomposition of <span class="math"><em>G</em></span> arises from the combination between graph decomposition and graph labeling. A decomposition of a graph <span class="math"><em>G</em></span> into isomorphic copies of a graph <span class="math"><em>H</em></span> is <span class="math"><em>H</em></span>-magic if there is a bijection <span class="math"><em>f</em> : <em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>) → {1, 2, ..., ∣<em>V</em>(<em>G</em>) ∪ <em>E</em>(<em>G</em>)∣}</span> such that the sum of labels of edges and vertices of each copy of <span class="math"><em>H</em></span> in the decomposition is constant. A lexicographic product of two graphs <span class="math"><em>G</em><sub>1</sub></span> and <span class="math"><em>G</em><sub>2</sub>, </span> denoted by <span class="math"><em>G</em><sub>1</sub>[<em>G</em><sub>2</sub>], </span> is a graph which arises from <span class="math"><em>G</em><sub>1</sub></span> by replacing each vertex of <span class="math"><em>G</em><sub>1</sub></span> by a copy of the <span class="math"><em>G</em><sub>2</sub></span> and each edge of <span class="math"><em>G</em><sub>1</sub></span> by all edges of the complete bipartite graph <span class="math"><em>K</em><sub><em>n</em>, <em>n</em></sub></span> where <span class="math"><em>n</em></span> is the order of <span class="math"><em>G</em><sub>2</sub>.</span> In this paper we provide a sufficient condition for <span class="math">$\overline{C_{n}}[\overline{K_{m}}]$</span> in order to have a <span class="math">$P_{t}[\overline{K_{m}}]$</span>-magic decompositions, where <span class="math"><em>n</em> &gt; 3, <em>m</em> &gt; 1, </span> and <span class="math"><em>t</em> = 3, 4, <em>n</em> − 2</span>.</p>

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