Let γ(G) and ir(G) denote the domination number and the irredundance number of a graph G, respectively. Allan and Laskar [1] and Bollobás and Cockayne [2] proved independently that γ(G) < 2 ir(G) for any graph G. For a tree T , Damaschke [4] obtained the sharper estimation 2γ(T ) < 3 ir(T ). Extending Damaschke’s result, Volkmann [11] proved that 2γ(G) ≤ 3 ir(G) for any block graph G and for an...

Golumbic, M.C. and R.C. Laskar, Irredundancy in circular arc graphs, Discrete Applied Mathematics 44 (1993) 79-89. A set ofvertices Xis called irredundant if for every x in Xthe closed neighborhood N[x] contains a vertex which is not a member of N[X-x], the union of the closed neighborhoods of the other vertices. In this paper we show that for circular arc graphs the size of the maximum irredun...

For a graph property P, in particular maximal independence, minimal domination and maximal irredundance, we introduce iterated P-colorings of graphs. The six graph parameters arising from either maximizing or minimizing the number of colors used for each property, are related by an inequality chain, and in this paper we initiate the study of these parameters. We relate them to other well-studie...

A set S of vertices in a graph G is called a total irredundant set if, for each vertex v in G; v or one of its neighbors has no neighbor in S −{v}. We investigate the minimum and maximum cardinalities of maximal total irredundant sets. c © 2002 Elsevier Science B.V. All rights reserved.

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...

A set D ⊆ V of vertices is said to be a (connected) distance k-dominating set of G if the distance between each vertex u ∈ V − D and D is at most k (and D induces a connected graph in G). The minimum cardinality of a (connected) distance k-dominating set in G is the (connected) distance k-domination number of G, denoted by γk(G) (γ c k (G), respectively). The set D is defined to be a total k-do...

Let β(G), Γ(G) and IR(G) be the independence number, the upper domination number and the upper irredundance number, respectively. A graph G is called Γperfect if β(H) = Γ(H), for every induced subgraph H of G. A graph G is called IR-perfect if Γ(H) = IR(H), for every induced subgraph H of G. In this paper, we present a characterization of Γ-perfect graphs in terms of some family of forbidden in...

A set D of vertices a graph G=(V,E) is irredundant if each non-isolated vertex G[D] has neighbour in V−D that not adjacent to any other D. The upper irredundance number IR(G) the largest cardinality an G; IR(G)-set IR(G). IR-graph G IR(G)-sets as set, and sets D′ are only can be obtained from by exchanging single for D′. An IR-tree tree. We characterize IR-trees diameter 3 showing these graphs ...

‎Let $G=(V(G),E(G))$ be a graph‎, ‎$gamma_t(G)$. Let $ooir(G)$ be the total domination and OO-irredundance number of $G$‎, ‎respectively‎. ‎A total dominating set $S$ of $G$ is called a $textit{total perfect code}$ if every vertex in $V(G)$ is adjacent to exactly one vertex of $S$‎. ‎In this paper‎, ‎we show that if $G$ has a total perfect code‎, ‎then $gamma_t(G)=ooir(G)$‎. ‎As a consequence, ...

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 ...