Further results on the total Italian domination number of trees
[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 verte...
| Autores: | , , |
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| Tipo de documento: | artigo |
| Data de publicação: | 2023 |
| País: | España |
| Recursos: | Universitat Politècnica de València (UPV) |
| Repositório: | RiuNet. Repositorio Institucional de la Universitat Politécnica de Valéncia |
| Idioma: | inglês |
| OAI Identifier: | oai:riunet.upv.es:10251/205853 |
| Acesso em linha: | https://riunet.upv.es/handle/10251/205853 |
| Access Level: | Acceso aberto |
| Palavra-chave: | Total Italian domination number Double domination number Domination number Trees ESTADISTICA E INVESTIGACION OPERATIVA |
| Resumo: | [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. |
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