scholarly journals γ-Graphs of Trees

Algorithms ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 153 ◽  
Author(s):  
Stephen Finbow ◽  
Christopher M. van Bommel
Keyword(s):  
Cartesian Product ◽  
Dominating Sets ◽  
Single Vertex ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Vertex Set ◽  
Cartesian Product Graphs ◽  
Product Of Graphs

For a graph G = ( V , E ) , the γ -graph of G, denoted G ( γ ) = ( V ( γ ) , E ( γ ) ) , is the graph whose vertex set is the collection of minimum dominating sets, or γ -sets of G, and two γ -sets are adjacent in G ( γ ) if they differ by a single vertex and the two different vertices are adjacent in G. In this paper, we consider γ -graphs of trees. We develop an algorithm for determining the γ -graph of a tree, characterize which trees are γ -graphs of trees, and further comment on the structure of γ -graphs of trees and its connections with Cartesian product graphs, the set of graphs which can be obtained from the Cartesian product of graphs of order at least two.

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 $A^n$.

Note on Reliability Analysis of Cartesian Product of Networks for Components

2021 ◽  
pp. 2142010
Author(s):  
Litao Guo ◽  
Jun Ge
Keyword(s):  
Reliability Analysis ◽  
Critical Parameter ◽  
Cartesian Product ◽  
Schwarz Inequality ◽  
Large Network ◽  
Vertex Number ◽  
Cartesian Product Of Graphs ◽  
Vertex Set ◽  
Product Of Graphs ◽  
Text Component

Connectivity is a critical parameter which can measure the reliability of networks. Let [Formula: see text] be a vertex set of [Formula: see text]. If [Formula: see text] has at least [Formula: see text] components, then [Formula: see text] is a [Formula: see text]-component cut of [Formula: see text]. The [Formula: see text]-component connectivity [Formula: see text] of [Formula: see text] is the vertex number of a smallest [Formula: see text]-component cut. Cartesian product of graphs is a useful method to construct a large network. We will use Cauchy–Schwarz inequality to determine the component connectivity of Cartesian product of some graphs.

scholarly journals On very strongly perfect Cartesian product graphs

2021 ◽  
Vol 7 (2) ◽  
pp. 2634-2645
Author(s):  
Ganesh Gandal ◽  
◽  
R Mary Jeya Jothi ◽  
Narayan Phadatare ◽  
Keyword(s):  
Cartesian Product ◽  
Sufficient Conditions ◽  
Perfect Graph ◽  
Finite Graphs ◽  
Cartesian Product Of Graphs ◽  
Product Graphs ◽  
Necessary And Sufficient ◽  
And Control ◽  
Product Of Graphs ◽  
Real Time Application

<abstract><p>Let $ G_1 \square G_2 $ be the Cartesian product of simple, connected and finite graphs $ G_1 $ and $ G_2 $. We give necessary and sufficient conditions for the Cartesian product of graphs to be very strongly perfect. Further, we introduce and characterize the co-strongly perfect graph. The very strongly perfect graph is implemented in the real-time application of a wireless sensor network to optimize the set of master nodes to communicate and control nodes placed in the field.</p></abstract>

scholarly journals Spectra of some Operations on Graphs

2016 ◽  
Vol 11 (9) ◽  
pp. 5654-5660
Author(s):  
Essam EI Seidy ◽  
Salah ElDin Hussein ◽  
Atef Abo Elkher
Keyword(s):  
Direct Product ◽  
Cartesian Product ◽  
Simple Graph ◽  
Simple Graphs ◽  
Cartesian Product Of Graphs ◽  
Vertex Set ◽  
Join Of Graphs ◽  
Product Of Graphs

In this paper, we consider a finite undirected and connected simple graph G(E, V) with vertex set V(G) and edge set E(G).We introduced a new computes the spectra of some operations on simple graphs [union of disjoint graphs, join of graphs, cartesian product of graphs, strong cartesian product of graphs, direct product of graphs].

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.

s-strongly perfect cartesian product of graphs

1992 ◽  
Vol 16 (4) ◽  
pp. 297-303
Author(s):  
Elefterie Olaru ◽  
Eugen M??ndrescu
Keyword(s):  
Cartesian Product ◽  
Cartesian Product Of Graphs ◽  
Product Of Graphs
Sign in / Sign up

Export Citation Format

Share Document