OJDM  Vol.5 No.3 , July 2015
On the Signed Domination Number of the Cartesian Product of Two Directed Cycles
Abstract: Let D be a finite simple directed graph with vertex set V(D) and arc set A(D). A function  is called a signed dominating function (SDF) if  for each vertex . The weight  of f is defined by . The signed domination number of a digraph D is . Let Cm × Cn denotes the cartesian product of directed cycles of length m and n. In this paper, we determine the exact values of gs(Cm × Cn) for m = 8, 9, 10 and arbitrary n. Also, we give the exact value of gs(Cm × Cn) when m,  (mod 3) and bounds for otherwise.
Cite this paper: Shaheen, R. (2015) On the Signed Domination Number of the Cartesian Product of Two Directed Cycles. Open Journal of Discrete Mathematics, 5, 54-64. doi: 10.4236/ojdm.2015.53005.

[1]   Haas, R. and Wexler, T.B. (1999) Bounds on the Signed Domination Number of a Graph. Discrete Mathematics, 195, 295-298.

[2]   West, D.B. (2000) Introduction to Graph Theory. Prentice Hall, Inc., Upper Saddle River.

[3]   Dunbar, J.E., Hedetniemi, S.T., Henning, M.A. and Slater, P.J. (1995) Signed Domination in Graphs, Graph Theory, Combinatorics and Application. John Wiley & Sons, Inc., Hoboken, 311-322.

[4]   Cockayne, E.J. and Mynhart, C.M. (1996) On a Generalization of Signed Domination Functions of Graphs. Ars Combinatoria, 43, 235-245.

[5]   Hattingh, J.H. and Ungerer, E. (1998) The Signed and Minus k-Subdomination Numbers of Comets. Discrete Mathematics, 183, 141-152.

[6]   Xu, B. (2001) On Signed Edge Domination Numbers of Graphs. Discrete Mathematics, 239, 179-189.

[7]   Broere, I., Hattingh, J.H., Henning, M.A. and McRae, A.A. (1995) Majority Domination in Graphs. Discrete Mathematics, 138, 125-135.

[8]   Zelinka, B. (2005) Signed Domination Numbers of Directed Graphs. Czechoslovak Mathematical Journal, 55, 479-482.

[9]   Karami, H., Sheikholeslami, S.M. and Khodkar, A. (2009) Lower Bounds on the Signed Domination Numbers of Directed Graphs. Discrete Mathematics, 309, 2567-2570.

[10]   Atapour, M., Sheikholeslami, S., Hajypory, R. and Volkmann, L. (2010) The Signed k-Domination Number of Directed Graphs. Central European Journal of Mathematics, 8, 1048-1057.

[11]   Shaheen, R. and Salim, H. (2015) The Signed Domination Number of Cartesian Products of Directed Cycles. Submitted to Utilitas Mathematica.