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 Cd1 × Cd2 for some special values of d, obtain that γd,4 (Cd1 × C3) = 2 if and only if d4(G) e1 ≤ d < d4(G) for d1 ≥ 5, γd,4 (Cd1 × C4) = 2 if d4(G) (2e1-[d1+e1]/2) ≤ d < d4(G) for d1 ≥ 24, and γd,4 (Cd1 × Cd2 ) = 2 if d4(G) ( e1-2) ≤ d < d4(G) for d1 = d2 ≥ 14.