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 $A^n$.

Bounds for complete cototal domination number of Cartesian product graphs and complement graphs

2021 ◽  
pp. 2150090
Author(s):  
J. Maria Regila Baby ◽  
K. Uma Samundesvari
Keyword(s):  
Dominating Set ◽  
Domination Number ◽  
Cartesian Product ◽  
Minimum Cardinality ◽  
Total Dominating Set ◽  
Product Graphs ◽  
Cartesian Product Graphs

A total dominating set [Formula: see text] is said to be a complete cototal dominating set if [Formula: see text] has no isolated nodes and it is represented by [Formula: see text]. The complete cototal domination number, represented by [Formula: see text], is the minimum cardinality of a [Formula: see text] set of [Formula: see text]. In this paper, the bounds for complete cototal domination number of Cartesian product graphs and complement graphs are determined.

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 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 Connected domination game played on Cartesian products

2019 ◽  
Vol 17 (1) ◽  
pp. 1269-1280 ◽  
Author(s):  
Csilla Bujtás ◽  
Pakanun Dokyeesun ◽  
Vesna Iršič ◽  
Sandi Klavžar
Keyword(s):  
Lower Bound ◽  
Domination Number ◽  
Cartesian Product ◽  
Connected Subgraph ◽  
Arbitrary Graph ◽  
Cartesian Products ◽  
Product Graphs ◽  
Cartesian Product Graphs ◽  
Game Domination Number ◽  
Domination Game

Abstract The connected domination game on a graph G is played by Dominator and Staller according to the rules of the standard domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of G. If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the connected game domination number of G. Here this invariant is studied on Cartesian product graphs. A general upper bound is proved and demonstrated to be sharp on Cartesian products of stars with paths or cycles. The connected game domination number is determined for Cartesian products of P3 with arbitrary paths or cycles, as well as for Cartesian products of an arbitrary graph with Kk for the cases when k is relatively large. A monotonicity theorem is proved for products with one complete factor. A sharp general lower bound on the connected game domination number of Cartesian products is also established.

scholarly journals Cartesian product graphs and k-tuple total domination

Filomat ◽  
2018 ◽  
Vol 32 (19) ◽  
pp. 6713-6731
Author(s):  
Adel Kazemi ◽  
Behnaz Pahlavsay ◽  
Rebecca Stones
Keyword(s):  
Dominating Set ◽  
Cartesian Product ◽  
Minimum Size ◽  
Constructive Proof ◽  
Extremal Case ◽  
Packing Number ◽  
Total Dominating Set ◽  
Product Graphs ◽  
Open Packing ◽  
Cartesian Product Graphs

A k-tuple total dominating set (kTDS) of a graph G is a set S of vertices in which every vertex in G is adjacent to at least k vertices in S; the minimum size of a kTDS is denoted ?xk,t(G). We give a Vizing-like inequality for Cartesian product graphs, namely ?xk,t(G) ?xk,t(H)? 2k?xk,t(G_H) provided ?xk,t(G) ? 2k?(G) holds, where ? denotes the packing number. We also give bounds on ?xk,t(G_H) in terms of (open) packing numbers, and consider the extremal case of ?xk,t(Kn_Km), i.e., the rook?s graph, giving a constructive proof of a general formula for ?x2,t(Kn_Km).

scholarly journals Connected domination number of cartesian product graphs of Cayley graphs with arithmetic graphs

2020 ◽  
Vol S (1) ◽  
pp. 48-51
Author(s):  
Uma Maheswari S. ◽  
Siva Parvathi M. ◽  
Bhatathi B. ◽  
Venkata Anusha M.
Keyword(s):  
Cayley Graphs ◽  
Domination Number ◽  
Cartesian Product ◽  
Product Graphs ◽  
Connected Domination Number ◽  
Cartesian Product Graphs
Sign in / Sign up

Export Citation Format

Share Document