scholarly journals Vizing-like Conjecture for the Upper Domination of Cartesian Products of Graphs – The Proof

10.37236/1979 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Boštjan Brešar
Keyword(s):  
Domination Number ◽  
Cartesian Product ◽  
Cartesian Products ◽  
Upper Domination Number ◽  
Arbitrary Graphs ◽  
Cartesian Products Of Graphs ◽  
Domination Numbers

In this note we prove the following conjecture of Nowakowski and Rall: For arbitrary graphs $G$ and $H$ the upper domination number of the Cartesian product $G \,\square \, H$ is at least the product of their upper domination numbers, in symbols: $\Gamma(G \,\square \, H)\ge \Gamma(G)\Gamma(H).$

scholarly journals Peg Solitaire on Cartesian Products of Graphs

2021 ◽  
Vol 37 (3) ◽  
pp. 907-917
Author(s):  
Martin Kreh ◽  
Jan-Hendrik de Wiljes
Keyword(s):  
Cartesian Product ◽  
Cartesian Products ◽  
Connected Graphs ◽  
Grid Graphs ◽  
Cartesian Products Of Graphs

AbstractIn 2011, Beeler and Hoilman generalized the game of peg solitaire to arbitrary connected graphs. In the same article, the authors proved some results on the solvability of Cartesian products, given solvable or distance 2-solvable graphs. We extend these results to Cartesian products of certain unsolvable graphs. In particular, we prove that ladders and grid graphs are solvable and, further, even the Cartesian product of two stars, which in a sense are the “most” unsolvable graphs.

scholarly journals A NOTE ON EDGE-CONNECTIVITY OF THE CARTESIAN PRODUCT OF GRAPHS

2011 ◽  
Vol 84 (1) ◽  
pp. 171-176
Author(s):  
LAKOA FITINA ◽  
C. T. LENARD ◽  
T. M. MILLS
Keyword(s):  
Cartesian Product ◽  
Edge Connectivity ◽  
Cartesian Products ◽  
Cartesian Product Of Graphs ◽  
Necessary And Sufficient ◽  
Product Of Graphs ◽  
Cartesian Products Of Graphs

AbstractThe main aim of this paper is to establish conditions that are necessary and sufficient for the edge-connectivity of the Cartesian product of two graphs to equal the sum of the edge-connectivities of the factors. The paper also clarifies an issue that has arisen in the literature on Cartesian products of graphs.

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 On the {k}-domination number of Cartesian products of graphs

2009 ◽  
Vol 309 (10) ◽  
pp. 3413-3419 ◽  
Author(s):  
Xinmin Hou ◽  
You Lu
Keyword(s):  
Domination Number ◽  
Cartesian Products ◽  
Cartesian Products Of Graphs

scholarly journals Bounds on the Size of the Minimum Dominating Sets of Some Cylindrical Grid Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-13
Author(s):  
Mrinal Nandi ◽  
Subrata Parui ◽  
Avishek Adhikari
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Domination Number ◽  
Cartesian Product ◽  
Wireless Sensor ◽  
Dominating Sets ◽  
Grid Graph ◽  
Grid Graphs ◽  
Cylindrical Grid ◽  
Domination Numbers

Let γPm □ Cn denote the domination number of the cylindrical grid graph formed by the Cartesian product of the graphs Pm, the path of length m, m≥2, and the graph Cn, the cycle of length n, n≥3. In this paper we propose methods to find the domination numbers of graphs of the form Pm □ Cn with n≥3 and m=5 and propose tight bounds on domination numbers of the graphs P6 □ Cn, n≥3. Moreover, we provide rough bounds on domination numbers of the graphs Pm □ Cn, n≥3 and m≥7. We also point out how domination numbers and minimum dominating sets are useful for wireless sensor networks.

scholarly journals Separation of Cartesian products of graphs into several connected components by the removal of edges

2021 ◽  
pp. 18-18
Author(s):  
Simon Spacapan
Keyword(s):  
Cartesian Product ◽  
Connected Components ◽  
Edge Connectivity ◽  
Cartesian Products ◽  
Minimum Cardinality ◽  
Connected Graphs ◽  
Cartesian Products Of Graphs

Let G = (V (G),E(G)) be a graph. A set S ? E(G) is an edge k-cut in G if the graph G-S = (V (G), E(G) \ S) has at least k connected components. The generalized k-edge connectivity of a graph G, denoted as ?k(G), is the minimum cardinality of an edge k-cut in G. In this article we determine generalized 3-edge connectivity of Cartesian product of connected graphs G and H and describe the structure of any minimum edge 3-cut in G2H. The generalized 3-edge connectivity ?3(G2H) is given in terms of ?3(G) and ?3(H) and in terms of other invariants of factors G and H.

scholarly journals More Results on Italian Domination in Cn□Cm

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 465 ◽  
Author(s):  
Hong Gao ◽  
Penghui Wang ◽  
Enmao Liu ◽  
Yuansheng Yang
Keyword(s):  
Domination Number ◽  
Cartesian Product ◽  
Cartesian Products ◽  
Rainbow Domination ◽  
Two Cities ◽  
Minimum Number ◽  
Graph Class ◽  
Domination In Graphs ◽  
Appl Math

Italian domination can be described such that in an empire all cities/vertices should be stationed with at most two troops. Every city having no troops must be adjacent to at least two cities with one troop or at least one city with two troops. In such an assignment, the minimum number of troops is the Italian domination number of the empire/graph is denoted as γ I . Determining the Italian domination number of a graph is a very popular topic. Li et al. obtained γ I ( C n □ C 3 ) and γ I ( C n □ C 4 ) (weak {2}-domination number of Cartesian products of cycles, J. Comb. Optim. 35 (2018): 75–85). Stȩpień et al. obtained γ I ( C n □ C 5 ) = 2 n (2-Rainbow domination number of C n □ C 5 , Discret. Appl. Math. 170 (2014): 113–116). In this paper, we study the Italian domination number of the Cartesian products of cycles C n □ C m for m ≥ 6 . For n ≡ 0 ( mod 3 ) , m ≡ 0 ( mod 3 ) , we obtain γ I ( C n □ C m ) = m n 3 . For other C n □ C m , we present a bound of γ I ( C n □ C m ) . Since for n = 6 k , m = 3 l or n = 3 k , m = 6 l ( k , l ≥ 1 ) , γ r 2 ( C n □ C m ) = m n 3 , (the Cartesian product of cycles with small 2-rainbow domination number, J. Comb. Optim. 30 (2015): 668–674), it follows in this case that C n □ C m is an example of a graph class for which γ I = γ r 2 , which can partially answer the question presented by Brešar et al. on the 2-rainbow domination in graphs, Discret. Appl. Math. 155 (2007): 2394–2400.

On the total {k}-domination number of Cartesian products of graphs

2008 ◽  
Vol 18 (2) ◽  
pp. 173-178 ◽  
Author(s):  
Ning Li ◽  
Xinmin Hou
Keyword(s):  
Domination Number ◽  
Cartesian Products ◽  
Cartesian Products Of Graphs

On the Total Domination Number of Cartesian Products of Graphs

2005 ◽  
Vol 21 (1) ◽  
pp. 63-69 ◽  
Author(s):  
Michael A. Henning ◽  
Douglas F. Rall
Keyword(s):  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Cartesian Products ◽  
Cartesian Products Of Graphs

scholarly journals Vizing's Conjecture for Graphs with Domination Number 3 - a New Proof

10.37236/5182 ◽  
2015 ◽  
Vol 22 (3) ◽  
Author(s):  
Boštjan Brešar
Keyword(s):  
Domination Number ◽  
Cartesian Product ◽  
Arbitrary Graph ◽  
Domination Numbers

Vizing's conjecture from 1968 asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. In this note we use a new, transparent approach to prove Vizing's conjecture for graphs with domination number 3;  that is, we prove that for any graph $G$ with $\gamma(G)=3$ and an arbitrary graph $H$, $\gamma(G\Box H) \ge 3\gamma(H)$.

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