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On a Conjecture About Signed Domination in the Cartesian Product of Two Directed Cycles

Abstract

Let D be a finite and simple digraph with vertex set V(D). A signed dominating function (SDF) of D is a function \(f:V(D)\longrightarrow \{-1,1\}\) such that \(f(N^{-}[v])=\sum _{x\in N^{-}[v]}f(x)\ge 1\) for every \(v\in V(D)\), where \(N^{-}[v]\) consists of v and all vertices of D from which arcs go into v. The weight of an SDF is the sum of its function values over all vertices, and the minimum weight of an SDF of G is the signed domination number \(\gamma _{s}(D).\) In this paper, we investigate the signed domination number of the Cartesian product of two directed cycles by showing that \(\gamma _{s}(C_{m}\Box C_{n})=\lceil \frac{m}{3}\rceil n\) if \(n\equiv 0\pmod {2m}\) or \(n\ge m\) and \(m\equiv 1\pmod 3,\) answering a conjecture posed in Shaheen (J Progress Res Math 6(2):770–777, 2016). Moreover, the exact value of \(\gamma _{s}(C_{8}\Box C_{n})\) is also provided.

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Correspondence to Seyed Mahmoud Sheikholeslami.

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Supported by the National Key Research and Development Program under Grant 2017YFB0802303, Applied Basic Research (Key Project) of Sichuan Province under Grant 2017JY0095, and the National Natural Science Foundation of China under Grants Nos. 61309015 and 61173121.

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Shao, Z., Jiang, H., Chellali, M. et al. On a Conjecture About Signed Domination in the Cartesian Product of Two Directed Cycles. Iran J Sci Technol Trans Sci 43, 2541–2549 (2019). https://doi.org/10.1007/s40995-019-00738-w

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Keywords

  • Signed domination number
  • Cartesian product
  • Cycle

Mathematics Subject Classification

  • 05C69