@article{Cabrera-Martínez2023, 
author = {Abel Cabrera-Martínez and Andrea Conchado Peiró and Juan Manuel Rueda-Vázquez},
title = {Further results on the total Italian domination number of trees},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {5},
pages = {10654-10664},
keywords = {total Italian domination number, double domination number, domination number, trees},
url = {https://www.sciopen.com/article/10.3934/math.2023540},
doi = {10.3934/math.2023540},
abstract = {Let    f  :  V  (  G  )  →  {  0  ,  1  ,  2  } be a function defined from a connected graph    G. Let        W    i    =  {  x  ∈  V  (  G  )  :  f  (  x  )  =  i  } for every    i  ∈  {  0  ,  1  ,  2  }. The function    f is called a total Italian dominating function on    G if        ∑          v      ∈      N      (      x      )        f  (  v  )  ≥  2 for every vertex    x  ∈      W    0   and if        ∑          v      ∈      N      (      x      )        f  (  v  )  ≥  1 for every vertex    x  ∈      W    1    ∪      W    2  . The total Italian domination number of    G, denoted by        γ          t      I        (  G  ), is the minimum weight    ω  (  f  )  =      ∑          x      ∈      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  γ  (  T  )  ≤      γ          t      I        (  T  )  ≤  n  (  T  )  −  γ  (  T  )  +  s  (  T  ),(ⅱ)              n      (      T      )      +      γ      (      T      )      +      s      (      T      )      −      l      (      T      )      +      1        2    ≤      γ          t      I        (  T  )  ≤            n      (      T      )      +      γ      (      T      )      +      l      (      T      )        2    ,where    γ  (  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.}
}