The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph in terms of order, minimum degree and m...

A dominating set S in a graph G is a tree dominating set of G if the subgraph induced by S is a tree. The tree domatic number of G is the maximum number of pairwise disjoint tree dominating sets in V (G). First, some exact values of and sharp bounds for the tree domatic number are given. Then, we establish a sharp lower bound for the number of edges in a connected graph of given order and given...

A signed Roman dominating function (SRDF) on a graph G is a function f : V (G) → {−1, 1, 2} such that u∈N [v] f(u) ≥ 1 for every v ∈ V (G), and every vertex u ∈ V (G) for which f(u) = −1 is adjacent to at least one vertex w for which f(w) = 2. A set {f1, f2, . . . , fd} of distinct signed Roman dominating functions on G with the property that ∑d i=1 fi(v) ≤ 1 for each v ∈ V (G), is called a sig...

Network lifetime is a critical issue in wireless sensor networks. In the coverage problem, sensors can be partitioned into many subsets to prolong network lifetime. These subsets are activated successively and each of them completely covers an interest region. Many centralized algorithms have been proposed to solve this problem. A very few distributed versions have also been presented but none ...

We study the problem of finding the smallest power of an input graph that has k disjoint dominating sets, where the ith power of an input graph G is constructed by adding edges between pairs of vertices in G at distance i or less, and a subset of vertices in a graph G is a dominating set if and only if every vertex in G is adjacent to a vertex in this subset. The problem is a different view of ...

We investigate the apparent difficulty of finding domatic partitions in graphs using tools from computability theory. We consider nicely presented (i.e., computable) infinite graphs and show that even if the domatic number is known, there might not be any algorithm for producing a domatic partition of optimal size. However, we prove that smaller domatic partitions can be constructed if we restr...

The paper studies the domatic numbers and the total domatic numbers of graphs having cut-vertices. We shall study the domatic number d(G) and the total domatic number d t (G) of a graph G. A survey of the related theory is given in 3]. We consider nite, undirected graphs without loops or multiple edges. A subset D of the vertex set V (G) of a graph G is called dominating (total dominating), if ...