The independent domination number of a graph G, denoted i(G), is the minimum cardinality of a maximal independent set of G. A maximal independent set of cardinality i(G) in G we call an i(G)-set. In this paper we provide a constructive characterization of trees G that have two disjoint i(G)-sets. © 2005 Elsevier B.V. All rights reserved.

The independent domination number of a graph G, denoted i(G), is the minimum cardinality of a maximal independent set of G. A maximal independent set of cardinality i(G) in G we call an i(G)-set. The graph G is called i-excellent if every vertex of G belongs to some i(G)-set. We provide a constructive characterization of i-excellent trees. c © 2002 Elsevier Science B.V. All rights reserved.

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...

The independent domination number of a graph is the smallest cardinality of an independent set that dominates the graph. In this paper we consider the independent domination number of triangle-free graphs. We improve several of the known bounds as a function of the order and minimum degree, thereby answering conjectures of Haviland.

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...

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 ...

Theaverage lower independencenumber iav(G)of a graphG=(V ,E) is defined as 1 |V | ∑ v∈V iv(G), and the average lower domination number av(G) is defined as 1 |V | ∑ v∈V v(G), where iv(G) (resp. v(G)) is the minimum cardinality of a maximal independent set (resp. dominating set) that contains v.We give an upper bound of iav(G) and av(G) for arbitrary graphs. Then we characterize the graphs achiev...

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 ...