A new upper bound for the perfect Italian domination number of a tree

A new upper bound for the bondage number of graphs with small domination number

Let, of the domination number of graph. The bondage number G was first introduced by Fink, Jacobson, the minimum among all sets of holds. In this paper we show that for graph G satisfying

A generalised upper bound for the k-tuple domination number

In this paper, we provide an upper bound for the k-tuple domination number that generalises known upper bounds for the double and triple domination numbers. We prove that for any graph G, ×k(G) ln( − k + 2)+ ln(∑k−1 m=1(k −m)d̂m + )+ 1 − k + 2 n, where ×k(G) is the k-tuple domination number; is the minimal degree; d̂m is the m-degree of G; = 1 if k = 1 or 2 and =−d if k 3; d is the average degree...

Lower bound on the paired domination number of a tree

Lower bound on the domination number of a tree

We prove that the domination number γ(T ) of a tree T on n ≥ 3 vertices and with n1 endvertices satisfies inequality γ(T ) ≥ n+2−n1 3 and we characterize the extremal graphs.

An Upper Bound on the Total Outer-independent Domination Number of a Tree

A total outer-independent dominating set of a graph G = (V (G), E(G)) is a set D of vertices of G such that every vertex of G has a neighbor in D, and the set V (G) \D is independent. The total outer-independent domination number of a graph G, denoted by γ t (G), is the minimum cardinality of a total outer-independent dominating set of G. We prove that for every tree T of order n ≥ 4, with l le...