For a positive integer k, a k-rainbow dominating function of a digraph D is a function f from the vertex set V (D) to the set of all subsets of the set {1, 2, . . . , k} such that for any vertex v ∈ V (D) with f(v) = ∅ the condition u∈N−(v) f(u) = {1, 2, . . . , k} is fulfilled, where N−(v) is the set of in-neighbors of v. A set {f1, f2, . . . , fd} of k-rainbow dominating functions on D with t...

Let. n ?: 1 be an integer and lei G = (V, E) be a graph. In this paper we study a non discrete generalization of l'n(G), the maximum cardinality of a minimal n-dominating sei in G. A real-valued function f : V -t [0,1] is n-dominating if for each v E V, the sum of the values assigned to the vertices in the closed n-neighbourhood of v, Nn[v], is at least one, i.e., f(Nn [ll]) ?: 1. The weight of...

This paper studies a nondiscrete generalization of T(G), the maximum cardinality of a minimal dominating set in a graph G = (K:E). In particular, a real-valued function f : V+ [0, l] is dominating if for each vertex DE V, the sum of the values assigned to the vertices in the closed neighborhood of u, N[o], is at least one, i.e., f (N[u]) 2 1. The weight of a dominating function f is f (V), the ...

For a graph $G=(V(G),E(G))$, an Italian dominating function (ID function) $f:V(G)\rightarrow\{0,1,2\}$ has the property that for every vertex $v\in V(G)$ with $f(v)=0$, either $v$ is adjacent to assigned $2$ under $f$ or least two vertices $1$ $f$. The weight of ID $\sum_{v\in V(G)}f(v)$. domination number minimum taken over all functions $G$. In this paper, we initiate study variant functions....

Let G = (V, E) be an undirected graph and r be a vertex weight function with positive integer values. A subset (clique) D ~_ V is an r-dominat ing set (clique) in G ifffor every vertex v e V there is a vertex u e D with dist(u, v) <~ r(v). This paper contains the following results: (i) We give a simple necessary and sufficient condition for the existence of r-dominating cliques in the case of H...

Let D be a simple digraph with vertex set V (D), and let f : V (D) → {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑x∈N−[v] f(x) ≥ k for each v ∈ V (D), where N[v] consists of v and all vertices of D from which arcs go into v, then f is a signed k-dominating function on D. A set {f1, f2, . . . , fd} of distinct signed k-dominating functions on D with the property that ∑d i=1 fi(x...

Efficient routing among mobile hosts is an important function in ad hoc networks. Routing based on a connected dominating set is a promising approach, where the search space for a route is reduced to the hosts in the set. A set is dominating if all the hosts are either in the set or neighbors of hosts in the set. The efficiency of dominating-set-based routing mainly depends on the overhead intr...

A number of optimization methods require as a rst step the construction of a dominating set (a set containing an optimal solution) enjoying properties such as compactness or convexity. In this note we address the problem of constructing dominating sets for problems whose objective is a componentwise nondecreasing function of (possibly an in nite number of) convex functions, and we show how to o...

A Roman dominating function on a graph G = (V,E) is a function f : V → {0, 1, 2} satisfying the condition that every vertex u with f(u) = 0 is adjacent to at least one vertex v with f(v) = 2. The weight of a Roman dominating function is the value f(G) = ∑ u∈V f(u). The Roman domination number of G is the minimum weight of a Roman dominating function on G. The Roman bondage number of a nonempty ...