On domination number of Cartesian product of directed cycles

2010 ◽  
Vol 110 (5) ◽  
pp. 171-173 ◽  
Author(s):  
Juan Liu ◽  
Xindong Zhang ◽  
Xing Chen ◽  
Jixiang Meng
Keyword(s):  
Domination Number ◽  
Cartesian Product ◽  
Directed Cycles

scholarly journals On the Domination Number of Cartesian Product of Two Directed Cycles

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zehui Shao ◽  
Enqiang Zhu ◽  
Fangnian Lang
Keyword(s):  
Upper Bound ◽  
Domination Number ◽  
Cartesian Product ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Dynamic Algorithm ◽  
Directed Cycles

Denote byγ(G)the domination number of a digraphGandCm□Cnthe Cartesian product ofCmandCn, the directed cycles of lengthm,n≥2. In 2010, Liu et al. determined the exact values ofγ(Cm□Cn)form=2,3,4,5,6. In 2013, Mollard determined the exact values ofγ(Cm□Cn)form=3k+2. In this paper, we give lower and upper bounds ofγ(Cm□Cn)withm=3k+1for different cases. In particular,⌈2k+1n/2⌉≤γ(C3k+1□Cn)≤⌊2k+1n/2⌋+k. Based on the established result, the exact values ofγ(Cm□Cn)are determined form=7and 10 by the combination of the dynamic algorithm, and an upper bound forγ(C13□Cn)is provided.

scholarly journals More Results on the Domination Number of Cartesian Product of Two Directed Cycles

Mathematics ◽  
2019 ◽  
Vol 7 (2) ◽  
pp. 210 ◽  
Author(s):  
Ansheng Ye ◽  
Fang Miao ◽  
Zehui Shao ◽  
Jia-Bao Liu ◽  
Janez Žerovnik ◽  
...  
Keyword(s):  
Graph Theory ◽  
Applied Mathematics ◽  
Domination Number ◽  
Cartesian Product ◽  
Equivalence Classes ◽  
Cartesian Products ◽  
Directed Cycles

Let γ ( D ) denote the domination number of a digraph D and let C m □ C n denote the Cartesian product of C m and C n , the directed cycles of length n ≥ m ≥ 3 . Liu et al. obtained the exact values of γ ( C m □ C n ) for m up to 6 [Domination number of Cartesian products of directed cycles, Inform. Process. Lett. 111 (2010) 36–39]. Shao et al. determined the exact values of γ ( C m □ C n ) for m = 6 , 7 [On the domination number of Cartesian product of two directed cycles, Journal of Applied Mathematics, Volume 2013, Article ID 619695]. Mollard obtained the exact values of γ ( C m □ C n ) for m = 3 k + 2 [M. Mollard, On domination of Cartesian product of directed cycles: Results for certain equivalence classes of lengths, Discuss. Math. Graph Theory 33(2) (2013) 387–394.]. In this paper, we extend the current known results on C m □ C n with m up to 21. Moreover, the exact values of γ ( C n □ C n ) with n up to 31 are determined.

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 $A^n$.

Non split hop domination number for some mirror graphs and cartesian product of two distinct paths

2018 ◽  
Vol 27 (2) ◽  
pp. 475-491
Author(s):  
G. Mahadevan ◽  
V. Vijayalakshmi ◽  
Selvam Avadayappan
Keyword(s):  
Domination Number ◽  
Cartesian Product
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