For a graph G = (V,E), a set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S as well as another vertex in V − S. The restrained domination number of G, denoted by γr(G), is the smallest cardinality of a restrained dominating set of G. In this paper we find all graphs G satisfying γr(G) = n− 3, where n is the order of G.

A set S of vertices of a graph G is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. We provide three equivalent conditions for a tree to have a unique minimum total dominating set and give a constructive characterization of such trees.

A set S of vertices of a graph G is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number t(G) is the minimum cardinality of a total dominating set of G. Let G be a connected spanning subgraph of Ks;s, and let H be the complement of G relative to Ks;s; that is, Ks;s = G ⊕ H is a factorization of Ks;s. The graph G is k-supercritical relative...

Electro-olfactogram recording was used to determine whether the olfactory epithelium of adult rainbow trout is specifically sensitive to bile acids, some of which have been hypothesized to function as pheromones. Of 38 bile acids that had been pre-screened for olfactory activity, 6 were selected. The rainbow trout-specific bile acids, taurocholic acid (TCA), and taurolithocholic acid 3-sulfate ...

Let G = (V,E) be a graph. A set S ⊆ V is a restrained dominating set if every vertex not in S is adjacent to a vertex in S and to a vertex in V \ S. The restrained domination number of G, denoted by γr(G), is the minimum cardinality of a restrained dominating set of G. A set S ⊆ V is a total dominating set if every vertex in V is adjacent to a vertex in S. The total domination number of a graph...

A total dominating set of a graph is a set of vertices such that every vertex is adjacent to a vertex in the set. We show that given a graph of order n with minimum degree at least 2, one can add at most (n−2√n )/4+O(log n) edges such that the resulting graph has two disjoint total dominating sets, and this bound is best possible.

We propose a memory efficient self-stabilizing protocol building k-independent dominating sets. A k-independent dominating set is a k-independent set and a kdominating set. A set of nodes, I, is k-independent if the distance between any pair of nodes in I is at least k + 1. A set of nodes, D, is a k-dominating if every node is within distance k of a node of D. Our algorithm, named SID, is silen...

A subset X of edges of a graph G is called an edge dominating set of G if every edge not in X is adjacent to some edge in X . The edge domination number γ′(G) of G is the minimum cardinality taken over all edge dominating sets of G. An edge Roman dominating function of a graph G is a function f : E(G) → {0, 1, 2} such that every edge e with f(e) = 0 is adjacent to some edge e′ with f(e′) = 2. T...

We propose a memory efficient self-stabilizing protocol building k-independent dominating sets. A k-independent dominating set is a k-independent set and a kdominating set. A set of nodes, I, is k-independent if the distance between any pair of nodes in I is at least k + 1. A set of nodes, D, is a k-dominating if every node is within distance k of a node of D. Our algorithm, named SID, is silen...