The Signed Domination Number of  Cartesian Products of Directed Cycles

Ramy Shaheen

Ramy Shaheen Department of Mathematics Faculty of Science, Tishreen University, Lattakia, Syria.

Keywords: Directed graph, Directed cycle, Cartesian product, Signed dominating function, Signed domination number.

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) iffor each vertex v. The weight w(f ) of f is defined by. The signed domination number of a digraph D is g_s(D) = min{w(f ) : f is an SDF of D}. Let C_m_n denotes the Cartesian product of directed cycles of length m and n. In this paper, we determine the exact value of signed domination number of some classes of Cartesian product of directed cycles. In particular, we prove that: (a) g_s(C₃_n) = n if n 0(mod 3), otherwise g_s(C₃_n) = n + 2. (b) g_s(C₄_n) = 2n. (c) g_s(C₅_n) = 2n if n 0(mod 10), g_s(C₅_n) = 2n + 1 if n 3, 5, 7(mod 10), g_s(C₅_n) = 2n + 2 if n 2, 4, 6, 8(mod 10), g_s(C₅_n) = 2n + 3 if n 1,9(mod 10). (d) g_s(C₆_n) = 2n if n 0(mod 3), otherwise g_s(C₆_n) = 2n + 4. (e) g_s(C₇_n) = 3n.

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References

Atapour M., Sheikholeslami S. M., Hajypory R. and Volkmann L., The signed k-domination number of directed graphs, Cent. Eur. J. Math., 8 (2010), 1048-1057.

Broere I., Hattingh J.H., M.A. Henning M.A., McRae A.A., Majority domination in graphs, Discrete Math., 138 (1995) 125–135.

Cockayne E.J., Mynhart C.M., On a generalization of signed domination functions of graphs, Ars. Combin., 43 (1996) 235–245.

Dunbar J.E., Hedetniemi S.T., Henning M.A., Slater P.J., Signed domination in graphs, Graph Theory, Combinatorics and Application, John Wiley & Sons, Inc., 1 (1995), 311–322.

Hass R., Wexler T.B., Bounds on the signed domination number of a graph, Discrete Math., 195 (1999), 295–298.

Hattingh J.H, Ungerer E., The signed and minus k-subdomination numbers of comets, Discrete Math., 183 (1998) 141–152.

Haynes T.W., S.T. Hedetniemi S.T., Slater P.J., Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998.

Karami H., Sheikholeslami S.M., Khodkar A., Lower bounds on the signed domination numbers of directed graphs, Discrete Math., 2009, 309(8), 2567–2570.

Shaheen R., On signed domination number of Cartesian Products of Directed Paths, Submitted.

Xu B., On signed edge domination numbers of graphs, Discrete Math., 239 (2001) 179–189.

Zelinka B., Signed domination numbers of directed graphs, Czechoslovak Math. J., 2005, 55(130)(2), 479–482.