Domination numbers of undirected toroidal mesh

Abstract

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 \(C_{d_1 } \times C_{d_2 }\) for some special values of d, obtain that \(\gamma _{d,4} \left( {C_{d_1 } \times C_3 } \right) = 2\) if and only if d4(G) − e1 ≤ d < d4(G) for d1 ≥ 5, \(\gamma _{d,4} \left( {C_{d_1 } \times C_4 } \right) = 2\) if \(d_4 \left( G \right) - \left( {2e_1 - \left\lfloor {\tfrac{{d_1 + e_1 }} {2}} \right\rfloor } \right) \leqslant d < d_4 \left( G \right)\) for d1 ≥ 24 and \(\gamma _{d,4} \left( {C_{d_1 } \times C_{d_2 } } \right) = 2\) if d4(G) − (e1 − 2) ≤ d < d4(G) for d1 = d2 ≥ 14.