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 *C*_{m} × *C*_{n} denotes the
cartesian product of directed cycles of length *m* and *n*. In this paper,
we determine the exact values of *g*_{s}(*C*_{m} × *C*_{n}) for *m* = 8, 9, 10 and arbitrary *n*. Also, we give the exact value of *g*_{s}(*C*_{m} × *C*_{n}) when *m*, (mod 3) and bounds for otherwise.

Let

KEYWORDS

Directed Graph, Directed Cycle, Cartesian Product, Signed Dominating Function, Signed Domination Number

Directed Graph, Directed Cycle, Cartesian Product, Signed Dominating Function, Signed Domination Number

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.

Shaheen, R. (2015) On the Signed Domination Number of the Cartesian Product of Two Directed Cycles.

References

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

http://dx.doi.org/10.1016/S0012-365X(98)00189-7

[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.

http://dx.doi.org/10.1016/S0012-365X(97)00051-4

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

http://dx.doi.org/10.1016/S0012-365X(01)00044-9

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

http://dx.doi.org/10.1016/0012-365X(94)00194-N

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

http://dx.doi.org/10.1007/s10587-005-0038-5

[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.

http://dx.doi.org/10.1016/j.disc.2008.04.001

[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.

http://dx.doi.org/10.2478/s11533-010-0077-5

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

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

http://dx.doi.org/10.1016/S0012-365X(98)00189-7

[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.

http://dx.doi.org/10.1016/S0012-365X(97)00051-4

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

http://dx.doi.org/10.1016/S0012-365X(01)00044-9

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

http://dx.doi.org/10.1016/0012-365X(94)00194-N

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

http://dx.doi.org/10.1007/s10587-005-0038-5

[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.

http://dx.doi.org/10.1016/j.disc.2008.04.001

[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.

http://dx.doi.org/10.2478/s11533-010-0077-5

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