An edge dominating set in a graph G is a set of edges D such that every edge not in D is adjacent to an edge of D. An edge domatic partition of a graph C=(V, E) is a collection of pairwise-disjoint edge dominating sets of G whose union is E. The maximum size of an edge domatic partition of G is called the edge domatic number. In this paper, we study the edge domatic number of the complete parti...

We show that the number of minimal dominating sets in a graph on n vertices is at most 1.7697, thus improving on the trivial O(2n/√n) bound. Our result makes use of the measure and conquer technique from exact algorithms, and can be easily turned into an O(1.7697) listing algorithm. Based on this result, we derive an O(2.8805n) algorithm for the domatic number problem, and an O(1.5780) algorith...

Roman dominating function} on a digraph $D$ with vertex set $V(D)$ is a labeling$fcolon V(D)to {0, 1, 2}$such that every vertex with label $0$ has an in-neighbor with label $2$. A set ${f_1,f_2,ldots,f_d}$ ofRoman dominating functions on $D$ with the property that $sum_{i=1}^d f_i(v)le 2$ for each $vin V(D)$,is called a {em Roman dominating family} (of functions) on $D$....

Let δ(G), ∆(G) and γ(G) be the minimum degree, maximum degree and domination number of a graph G = (V (G), E(G)), respectively. A partition of V (G), all of whose classes are dominating sets in G, is called a domatic partition of G. The maximum number of classes of a domatic partition of G is called the domatic number of G, denoted d(G). It is well known that d(G) ≤ δ(G)+1, d(G)γ(G) ≤ |V (G)| [...

Let k be a positive integer. A subset S of V (G) in a graph G is a k-tuple total dominating set of G if every vertex of G has at least k neighbors in S. The k-tuple total domination number γ×k,t(G) of G is the minimum cardinality of a k-tuple total dominating set of G. In this paper for a given graph G with minimum degree at least k, we find some sharp lower and upper bounds on the k-tuple tota...

The domatic number d(G) of a graph G = (V,E) is the maximum order of a partition of V into dominating sets. Such a partition Π = {D1, D2, . . . , Dd} is called a minimal dominating d-partition if Π contains the maximum number of minimal dominating sets, where the maximum is taken over all d-partitions of G. The minimal dominating d-partition number Λ(G) is the number of minimal dominating sets ...