scholarly journals Bounds on the domination number and the metric dimension of co-normal product of graphs

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Imran Javaid ◽  
Shahid ur Rehman ◽  
Muhammad Imran
Keyword(s):  
Domination Number ◽  
Metric Dimension ◽  
Normal Product ◽  
Product Of Graphs

scholarly journals A Note on Total and Paired Domination of Cartesian Product Graphs

10.37236/2535 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
K. Choudhary ◽  
S. Margulies ◽  
I. V. Hicks
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Minimum Dominating Set ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Paired Domination ◽  
Product Of Graphs

A dominating set $D$ for a graph $G$ is a subset of $V(G)$ such that any vertex not in $D$ has at least one neighbor in $D$. The domination number $\gamma(G)$ is the size of a minimum dominating set in G. Vizing's conjecture from 1968 states that for the Cartesian product of graphs $G$ and $H$, $\gamma(G)\gamma(H) \leq \gamma(G \Box H)$, and Clark and Suen (2000) proved that $\gamma(G)\gamma(H) \leq 2 \gamma(G \Box H)$. In this paper, we modify the approach of Clark and Suen to prove a variety of similar bounds related to total and paired domination, and also extend these bounds to the $n$-Cartesian product of graphs $A^1$ through $A^n$.

scholarly journals Inequality Related to Vizing's Conjecture

10.37236/1542 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
W. Edwin Clark ◽  
Stephen Suen
Keyword(s):  
Domination Number ◽  
Cartesian Product ◽  
Simple Graphs ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs

Let $\gamma(G)$ denote the domination number of a graph $G$ and let $G\square H$ denote the Cartesian product of graphs $G$ and $H$. We prove that $\gamma(G)\gamma(H) \le 2 \gamma(G\square H)$ for all simple graphs $G$ and $H$.

scholarly journals On resolvability of a graph associated to a finite vector space

2019 ◽  
Vol 18 (02) ◽  
pp. 1950029
Author(s):  
U. Ali ◽  
S. A. Bokhary ◽  
K. Wahid ◽  
G. Abbas
Keyword(s):  
Vector Space ◽  
Domination Number ◽  
Exchange Property ◽  
Metric Dimension ◽  
Resolving Set ◽  
Matroid Intersection ◽  
Finite Vector ◽  
Finite Vector Space ◽  
Partition Dimension ◽  
Matroid Intersection Problem

In this paper, the resolving parameters such as metric dimension and partition dimension for the nonzero component graph, associated to a finite vector space, are discussed. The exact values of these parameters are determined. It is derived that the notions of metric dimension and locating-domination number coincide in the graph. Independent sets, introduced by Boutin [Determining sets, resolving set, and the exchange property, Graphs Combin. 25 (2009) 789–806], are studied in the graph. It is shown that the exchange property holds in the graph for minimal resolving sets with some exceptions. Consequently, a minimal resolving set of the graph is a basis for a matroid with the set [Formula: see text] of nonzero vectors of the vector space as the ground set. The matroid intersection problem for two matroids with [Formula: see text] as the ground set is also solved.

scholarly journals On the Locating Edge Domination Number of Comb Product of Graphs

2018 ◽  
Vol 1022 ◽  
pp. 012003 ◽  
Author(s):  
Dafik ◽  
I. H. Agustin ◽  
Moh. Hasan ◽  
R. Adawiyah ◽  
R. Alfarisi ◽  
...  
Keyword(s):  
Domination Number ◽  
Product Of Graphs

scholarly journals The metric dimension of the lexicographic product of graphs

2013 ◽  
Vol 313 (9) ◽  
pp. 1045-1051 ◽  
Author(s):  
S.W. Saputro ◽  
R. Simanjuntak ◽  
S. Uttunggadewa ◽  
H. Assiyatun ◽  
E.T. Baskoro ◽  
...  
Keyword(s):  
Metric Dimension ◽  
Lexicographic Product ◽  
Product Of Graphs

scholarly journals New results on connected dominating structures in graphs

2019 ◽  
Vol 11 (1) ◽  
pp. 52-64
Author(s):  
Libin Chacko Samuel ◽  
Mayamma Joseph
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Forbidden Subgraph ◽  
Dominating Sets ◽  
Minimum Cardinality ◽  
Complete Subgraph ◽  
Strong Product ◽  
Necessary And Sufficient ◽  
Domination In Graphs ◽  
Product Of Graphs

Abstract A set of vertices in a graph is a dominating set if every vertex not in the set is adjacent to at least one vertex in the set. A dominating structure is a subgraph induced by the dominating set. Connected domination is a type of domination where the dominating structure is connected. Clique domination is a type of domination in graphs where the dominating structure is a complete subgraph. The clique domination number of a graph G denoted by γk(G) is the minimum cardinality among all the clique dominating sets of G. We present few properties of graphs admitting dominating cliques along with bounds on clique domination number in terms of order and size of the graph. A necessary and sufficient condition for the existence of dominating clique in strong product of graphs is presented. A forbidden subgraph condition necessary to imply the existence of a connected dominating set of size four also is found.

scholarly journals Forcing Total dr-Power Domination Number of Graphs Under Some Binary Operations

2021 ◽  
Vol 14 (3) ◽  
pp. 1098-1107
Author(s):  
Cris Laquibla Armada
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Domination Number ◽  
Lexicographic Product ◽  
Power Domination ◽  
Binary Operations ◽  
Product Of Graphs

In this paper, the total dr-power domination number of graphs such as complete bipartite graph, generalized fan and generalized wheel are obtained. The forcing total dr-power domination number of graphs resulting from some binary operations such as join, corona and lexicographic product of graphs were determined.

Sign in / Sign up

Export Citation Format

Share Document