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Discussiones Mathematicae Graph Theory 33(2) (2013)
387-394
DOI: https://doi.org/10.7151/dmgt.1668
On the domination of Cartesian product of directed cycles: Results for certain equivalence classes of lengths
| Michel Mollard
CNRS Université Joseph Fourier |
Abstract
Let γ(Cm→□Cn→) be the domination number of the Cartesian product of directed cycles Cm→ and Cn→ for m,n ≥ 2. Shaheen [13] and Liu et al. ([11], [12]) determined the value of γ(Cm→□Cn→) when m ≤ 6 and [12] when both m and n ≡ 0 (mod 3). In this article we give, in general, the value of γ(Cm→□Cn→) when m ≡ 2(mod 3) and improve the known lower bounds for most of the remaining cases. We also disprove the conjectured formula for the case m ≡ 0(mod 3) appearing in [12].
Keywords: directed graph, Cartesian product, domination number, directed cycle
2010 Mathematics Subject Classification: 05C69,05C38.
References
| [1] | T.Y. Chang and W.E. Clark, The domination numbers of the 5 × n and 6 × n grid graphs, J. Graph Theory 17 (1993) 81--107, doi: 10.1002/jgt.3190170110. |
| [2] | M. El-Zahar and C.M. Pareek, Domination number of products of graphs, Ars Combin. 31 (1991) 223--227. |
| [3] | M. El-Zahar, S. Khamis and Kh. Nazzal, On the domination number of the Cartesian product of the cycle of length n and any graph, Discrete Appl. Math. 155 (2007) 515--522, doi: 10.1016/j.dam.2006.07.003. |
| [4] | R.J. Faudree and R.H. Schelp, The domination number for the product of graphs, Congr. Numer. 79 (1990) 29--33. |
| [5] | S. Gravier and M. Mollard, On domination numbers of Cartesian product of paths, Discrete Appl. Math. 80 (1997) 247--250, doi: 10.1016/S0166-218X(97)00091-7. |
| [6] | B. Hartnell and D. Rall, On dominating the Cartesian product of a graph and K2, Discuss. Math. Graph Theory 24 (2004) 389--402, doi: 10.7151/dmgt.1238. |
| [7] | T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs ( Marcel Dekker, Inc. New York, 1998). |
| [8] | T.W. Haynes, S.T. Hedetniemi and P.J. Slater , Domination in Graphs: Advanced Topics (Marcel Dekker, Inc. New York, 1998). |
| [9] | M.S. Jacobson and L.F. Kinch, On the domination number of products of graphs I, Ars Combin. 18 (1983) 33--44. |
| [10] | S. Klavžar and N. Seifter, Dominating Cartesian products of cycles, Discrete Appl. Math. 59 (1995) 129--136, doi: 10.1016/0166-218X(93)E0167-W. |
| [11] | J. Liu, X.D. Zhang, X. Chenand and J. Meng, On domination number of Cartesian product of directed cycles, Inform. Process. Lett. 110 (2010) 171--173, doi: 10.1016/j.ipl.2009.11.005. |
| [12] | J. Liu, X.D. Zhang, X. Chen and J. Meng, Domination number of Cartesian products of directed cycles, Inform. Process. Lett. 111 (2010) 36--39, doi: 10.1016/j.ipl.2010.10.001. |
| [13] | R.S. Shaheen, Domination number of toroidal grid digraphs, Util. Math. 78 (2009) 175--184. |
Received 8 April 2011
Revised 24 May 2012
Accepted 28 May 2012
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