Let $G=(V,E)$ be a finite and simple graph of order $n$ maximumdegree $\Delta$. A signed strong total Roman dominating function ona $G$ is $f:V(G)\rightarrow\{-1, 1,2,\ldots, \lceil\frac{\Delta}{2}\rceil+1\}$ satisfying the condition that (i) forevery vertex $v$ $G$, $f(N(v))=\sum_{u\in N(v)}f(u)\geq 1$, where$N(v)$ open neighborhood (ii) every forwhich $f(v)=-1$ adjacent to at least one vertex...

Let D be a finite and 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 of D with the property tha...

Let G be a graph with vertex set V (G), and let f : V (G) −→ {−1, 1} be a two-valued function. If k ≥ 1 is an integer and ∑ x∈N(v) f(x) ≥ k for each v ∈ V (G), where N(v) is the neighborhood of v, then f is a signed total k-dominating function on G. A set {f1, f2, . . . , fd} of distinct signed total k-dominating functions on G with the property that ∑d i=1 fi(x) ≤ k for each x ∈ V (G), is call...

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 ...

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...