Domination numbers of undirected toroidal mesh

The (d,m)-domination number γd,m is a new measure to characterize the reliability of resources-sharing in fault tolerant networks, in some sense, which can more accurately characterize the reliability of networks than the m-diameter does. In this paper, we study the (d, 4)-domination numbers of undirected toroidal mesh \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{d_1 } \times C_{d_2 }$$\end{document} for some special values of d, obtain that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{d,4} \left( {C_{d_1 } \times C_3 } \right) = 2$$\end{document} if and only if d4(G) − e1 ≤ d < d4(G) for d1 ≥ 5, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{d,4} \left( {C_{d_1 } \times C_4 } \right) = 2$$\end{document} if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_4 \left( G \right) - \left( {2e_1 - \left\lfloor {\tfrac{{d_1 + e_1 }} {2}} \right\rfloor } \right) \leqslant d < d_4 \left( G \right)$$\end{document} for d1 ≥ 24 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma _{d,4} \left( {C_{d_1 } \times C_{d_2 } } \right) = 2$$\end{document} if d4(G) − (e1 − 2) ≤ d < d4(G) for d1 = d2 ≥ 14.