no edges does not have a total dominating set. The strong domination number of is defined as: ; obviously, . A dominating edge-set is a set of edges (vertex pairs) whose union is a dominating set; such a set may not exist (for example, a graph with one or more vertices and no edges does not have it). If it exists, then the union of all its edges is a strongly-dominating set. Therefore, the smallest size of an edge-dominating set is at least . In contrast, an edge-dominating set is a set D of edges, such that every edge not in D is adjacent to at least one edge in D; such a set always exists (for example, the set of all edges is an edge-dominating set). A k-dominating set is a set of vertices such that each vertex not in the set has at least k neighbors in the set (a standard dominating set is a 1-dominating set). Similarly, a k''-tuple dominating set is a set of vertices such that each vertex in the graph has at least k neighbors in the set (a total dominating set is a 1-tuple dominating set). An -approximation of a minimum k-tuple dominating set can be found in polynomial time. Every graph admits a k-dominating set (for example, the set of all vertices); but only graphs with minimum degree admit a k-tuple dominating set. However, even if the graph admits k-tuple dominating set, a minimum k-tuple dominating set can be nearly k times as large as a minimum k-dominating set for the same graph; An -approximation of a minimum k-dominating set can be found in polynomial time as well. A star-dominating set is a subset D of V such that, for every vertex v in V, the star of v (the set of edges adjacent to v) intersects the star of some vertex in D. Clearly, if G has isolated vertices then it has no star-dominating sets (since the star of isolated vertices is empty). If G has no isolated vertices, then every dominating set is a star-dominating set and vice versa. The distinction between star-domination and usual domination is more substantial when their fractional variants are considered. A domatic partition is a partition of the vertices into disjoint dominating sets. The domatic number is the maximum size of a domatic partition. An eternal dominating set is a dynamic version of domination in