It is shown that the lower irredundance number and secure domination number of an n vertex tree T with maximum degree 3, are bounded below by 2(n+ 1)/(2 + 3) (T = K1, ) and ( n+ − 1)/(3 − 1), respectively. The bounds are sharp and extremal trees are exhibited. © 2006 Elsevier B.V. All rights reserved. MSC: 05C69

This paper investigates cases where one graph parameter, upper fractional domination, is equal to three others: independence, upper domination and upper irredundance. We show that they are all equal for a large subclass, known as strongly perfect graphs, of the class of perfect graphs. They are also equal for odd cycles and upper bound graphs. However for simplicial graphs, upper irredundance m...

Let (G) and IR(G) denote the independence number and the upper irredundance number of a graph G. We prove that in any graph of order n, minimum degree and maximum degree =0, IR(G)6 n=(1 + = ) and IR(G) − (G)6 (( − 2)=2 )n. The two bounds are attained by arbitrarily large graphs. The second one proves a conjecture by Rautenbach related to the case = 3. When the chromatic number of G is less than...

Let ir(G) and γ(G) be the irredundance number and domination number of a graph G, respectively. The number of vertices and leafs of a graph G is denoted by n(G) and n1(G). If T is a tree, then Lemańska [4] presented in 2004 the sharp lower bound γ(T ) ≥ n(T ) + 2− n1(T ) 3 . In this paper we prove ir(T ) ≥ n(T ) + 2− n1(T ) 3 for an arbitrary tree T . Since γ(T ) ≥ ir(T ) is always valid, this ...

In this article we present characterizations of locally well-dominated graphs and locally independent well-dominated graphs, and a sufficient condition for a graph to be k-locally independent well-dominated. Using these results we show that the irredundance number, the domination number and the independent domination number can be computed in polynomial time within several classes of graphs, e....

This paper contains a number of estimations of the split domination number and the maximal domination number of a graph with a deleted subset of edges which induces a complete subgraph Kp. We discuss noncomplete graphs having or not having hanging vertices. In particular, for p = 2 the edge deleted graphs are considered. The motivation of these problems comes from [2] and [6], where the authors...