Let k be a positive integer and G be a connected graph. This paper considers the relations among four graph theoretical parameters: the k-domination number k(G), the connected k-domination number c k (G); the k-independent domination number i k (G) and the k-irredundance number irk(G). The authors prove that if an irk-set X is a k-independent set of G, then irk(G) = k(G) = k(G), and that for k ...

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γtr(G), is the smallest cardinality of a total restrained dominating set of G. We show that if T is a tree of order n, then γtr(T ) ≥ d(n + 2)/2e. Moreover, we s...

In this paper, we consider graphs having a unique minimum independent dominating set. We first discuss the effects of deleting a vertex, or the closed neighborhood of a vertex, from such graphs. We then discuss five operations which, in certain circumstances, can be used to combine two graphs, each having a unique minimum independent dominating set, to produce a new graph also having a unique m...

A graph G is 3-domination-critical (3-critical, for short), if its domination number γ is 3 and the addition of any edge decreases γ by 1. In this paper, we show that every 3-critical graph with independence number 4 and minimum degree 3 is Hamilton-connected. Combining the result with those in [2], [4] and [5], we solve the following conjecture: a connected 3critical graph G is Hamilton-connec...

For a graphical property P and a graph G, we say that a subset S of the vertices of G is a P-set if the subgraph induced by S has the property P. Then the P-domination number of G is the minimum cardinality of a dominating P-set and the P-independence number the maximum cardinality of a P-set. We show that several properties of domination, independent domination and acyclic domination hold for ...

The following inequality chain has been extensively studied in the discrete mathematical literature: i r ~ y ~ i ~ f l ~ F ~IR, where ir and IR denote the lower and upper irredundance numbers of a graph, 2: and F denote the lower and upper domination numbers of a graph, i denotes the independent domination number and fl denotes the vertex independence number of a graph. More than one hundred pa...

In this paper we present upper bounds on the differences between the independence, domination and irredundance parameters of a graph. For example, using the Brooks theorem on the chromatic number, we show that for any graph G of order n with maximum degree ∆ ≥ 2 IR(G)− β(G) ≤ ⌊ ∆− 2 2∆ n ⌋ , where β(G) and IR(G) are the independence number and the upper irredundance number of a graph G, respect...

Given a graph G together with a capacity function c : V (G) → N, we call S ⊆ V (G) a capacitated dominating set if there exists a mapping f : (V (G) \ S) → S which maps every vertex in (V (G) \S) to one of its neighbors such that the total number of vertices mapped by f to any vertex v ∈ S does not exceed c(v). In the Planar Capacitated Dominating Set problem we are given a planar graph G, a ca...

We study a function on graphs, denoted by Gamma , representing vectorially the domination number of a graph, in a way similar to that in which the Lovász Theta function represents the independence number of a graph. This function is a lower bound on the homological connectivity of the independence complex of the graph, and hence is of value in studying matching problems by topological methods. ...

A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. A fundamental problem in total domination theory in graphs is to determine which graphs have two disjoint total dominating sets. In this paper, we solve this problem and provide a constructive characterization of the graphs that have two disjoint total dominating sets.