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
For a simple digraph G, let β（G） be the size of the smallest subset X E（G） such that G - X has no directed cycles, and let γ（G） be the number of unordered pairs of nonadjacent vertices in G. A digraph G is called k-free if G has no directed cycles of length at most k. This paper proves that β（G） ≤ 0.3819γ（G） if G is a 4-free digraph, and β（G） ≤ 0.2679γ（G） if G is a 5-free digraph. These improve the results of Sullivan in 2008.
A graph G satisfies the Ore-condition if d（x）＋d（y） ≥ ｜V（G）｜ for any xy E（G）. Luo et al. [European J. Combin., 2008] characterized the simple Z3-connected graphs satisfying the Ore-condition. In this paper, we characterize the simple Z3-connected graphs G satisfying d（x）＋d（y） ≥ ｜V（G）｜ - 1 for any xy E（G）, which improves the results of Luo et al.
