The Signed Domination Number of  Cartesian Products of Directed Cycles

Ramy Shaheen; Hussein Salim

The Signed Domination Number of Cartesian Products of Directed Cycles

Ramy Shaheen, Hussein Salim

Abstract

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.

Keywords

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

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ISSN: 2395-0218.

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