Abstract
Let γ(P m □P n ) be the domination number of the Cartesian product of directed paths P m and P n for m,n≥2. Liu et al. in (J. Comb. Optim. 22(4):651–662, 2011) determined the value of γ(P m □P n ) for arbitrary n and m≤6. In this work we give the exact value of γ(P m □P n ) for any m,n and exhibit dominating sets of minimum cardinality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chang TY (1992) Domination numbers of grid graphs. PhD thesis, University of South Florida, Tampa, FL
Chang TY, Clark WE (1993) The domination numbers of the 5×n and 6×n grid graphs. J Graph Theory 17:81–107
El-Zahar M, Pareek CM (1991) Domination number of products of graphs. Ars Comb 31:223–227
El-Zahar M, Khamis S, Nazzal Kh (2007) On the domination number of the Cartesian product of the cycle of length n and any graph. Discrete Appl Math 155:515–522
Faudree RJ, Schelp RH (1990) The domination number for the product of graphs. Congr Numer 79:29–33
Gonçalves D, Pinlou A, Rao M, Thomassé S (2011) The domination number of grids. SIAM J Discrete Math 25(3):1443–1453
Gravier S, Mollard M (1997) On domination numbers of Cartesian product of paths. Discrete Appl Math 80:247–250
Hartnell B, Rall D (2004) On dominating the Cartesian product of a graph and K 2. Discuss Math Graph Theory 24(3):389–402
Haynes TW, Hedetniemi ST, Slater PJ (1998a) Fundamentals of domination in graphs. Marcel Dekker, New York
Haynes TW, Hedetniemi ST, Slater PJ (eds.) (1998b) Domination in graphs: advanced topics. Marcel Dekker, New York
Jacobson MS, Kinch LF (1983) On the domination number of products of graphs I. Ars Comb 18:33–44
Klavžar S, Seifter N (1995) Dominating Cartesian products of cycles. Discrete Appl Math 59:129–136
Liu J, Zhang XD, Chen X, Meng J (2010) On domination number of Cartesian products of directed cycles. Inf Process Lett 110(5):171–173
Liu J, Zhang XD, Meng J (2011) On domination number of Cartesian product of directed paths. J Comb Optim 22(4):651–662
Mollard M (2011) On domination of Cartesian product of directed cycles. Submitted. Manuscript available on-line: http://hal.archives-ouvertes.fr/hal-00576481/fr/
Shaheen RS (2009) Domination number of toroidal grid digraphs. Util Math 78:175–184
Zhang XD, Liu J, Chen X, Meng J (2010) Domination number of Cartesian products of directed cycles. Inf Process Lett 111(1):36–39
Author information
Authors and Affiliations
Corresponding author
Additional information
CNRS Université Joseph Fourier.
Rights and permissions
About this article
Cite this article
Mollard, M. The domination number of Cartesian product of two directed paths. J Comb Optim 27, 144–151 (2014). https://doi.org/10.1007/s10878-012-9494-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-012-9494-7