On the Domination of Cartesian Product of Directed Cycles: Results for Certain Equivalence Classes of Lengths

Michel Mollard

Discussiones Mathematicae Graph Theory (2013)

  • Volume: 33, Issue: 2, page 387-394
  • ISSN: 2083-5892

Abstract

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Let (−→ Cm2−→ 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 (−→ Cm2−→ Cn) when m ≤ 6 and [12] when both m and n ≡ 0(mod 3). In this article we give, in general, the value of (−→ Cm2−→ 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].

How to cite

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Michel Mollard. "On the Domination of Cartesian Product of Directed Cycles: Results for Certain Equivalence Classes of Lengths." Discussiones Mathematicae Graph Theory 33.2 (2013): 387-394. <http://eudml.org/doc/267582>.

@article{MichelMollard2013,
abstract = {Let (−→ Cm2−→ 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 (−→ Cm2−→ Cn) when m ≤ 6 and [12] when both m and n ≡ 0(mod 3). In this article we give, in general, the value of (−→ Cm2−→ 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].},
author = {Michel Mollard},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {directed graph; Cartesian product; domination number; directed cycle},
language = {eng},
number = {2},
pages = {387-394},
title = {On the Domination of Cartesian Product of Directed Cycles: Results for Certain Equivalence Classes of Lengths},
url = {http://eudml.org/doc/267582},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Michel Mollard
TI - On the Domination of Cartesian Product of Directed Cycles: Results for Certain Equivalence Classes of Lengths
JO - Discussiones Mathematicae Graph Theory
PY - 2013
VL - 33
IS - 2
SP - 387
EP - 394
AB - Let (−→ Cm2−→ 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 (−→ Cm2−→ Cn) when m ≤ 6 and [12] when both m and n ≡ 0(mod 3). In this article we give, in general, the value of (−→ Cm2−→ 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].
LA - eng
KW - directed graph; Cartesian product; domination number; directed cycle
UR - http://eudml.org/doc/267582
ER -

References

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  7. [7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc. New York, 1998). Zbl0890.05002
  8. [8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs: Advanced Topics (Marcel Dekker, Inc. New York, 1998). Zbl0883.00011
  9. [9] M.S. Jacobson and L.F. Kinch, On the domination number of products of graphs I , Ars Combin. 18 (1983) 33-44. Zbl0566.05050
  10. [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[Crossref] Zbl0824.05037
  11. [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[Crossref] 
  12. [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[Crossref] Zbl1259.05134
  13. [13] R.S. Shaheen, Domination number of toroidal grid digraphs, Util. Math. 78 (2009) 175-184. Zbl1284.05203

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