Further results on the total Italian domination number of trees

Abstract

[EN] Let f : V(G) -> {0, 1, 2} be a function defined from a connected graph G. Let W-i = {x is an element of V(G) : f(x) = i} for every i is an element of{0, 1, 2}. The function f is called a total Italian dominating function on G if Z(v is an element of N(x)) f(v) >= 2 for every vertex x is an element of W-0 and if Z(v is an element of N(x)) f(v) >= 1 for every vertex x is an element of W-1 boolean OR W-2. The total Italian domination number of G, denoted by gamma(tI)(G), is the minimum weight omega(f) = Sigma(x is an element of V(G)) f (x) among all total Italian dominating functions f on G. In this paper, we provide new lower and upper bounds on the total Italian domination number of trees. In particular, we show that if T is a tree of order n(T) >= 2, then the following inequality chains are satisfied.

(i) 2 gamma(T) <= gamma(tI)(T) <= n(T) - gamma(T) + s(T),

(ii) n(T)+gamma(T)+s(T)-l(T )+1/2 <= gamma(tI)(T) <= n(T)+gamma(T)+l(T)/2 where gamma(T), s(T) and l(T) represent the classical domination number, the number of support vertices and the number of leaves of T, respectively. The upper bounds are derived from results obtained for the double domination number of a tree.