For general graphs, the t-LBTSS problem is not only NP-hard, it is also hard to be approximated by Chen’s inapproachability results (Chen, 2009). In this paper, we are interested in finding an optimal target set for some special family of graphs. A simple, tight but nontrivial inequality was presented which gives the lower bound of the total sum of degrees in a feasible target set to t-LBTSS problem, in terms of the number of edges in the graph. Necessary and sufficient conditions for equality to hold have been established, based on which we are able to construct families of infinite number of graphs for which the optimal solution to t-LBTSS problem become obvious. In particular, we gave an exact formula for the optimal solution of a kind of toroidal mesh graphs, while it seems difficult to tell what the optimal solutions are for these graphs without using the equality given in the paper.