Total restrained domination in trees

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Total restrained domination in trees

Let G = (V,E) be a graph. A set S ⊆ V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γtr(G), is the smallest cardinality of a total restrained dominating set of G. We show that if T is a tree of order n, then γtr(T ) ≥ d(n + 2)/2e. Moreover, we s...

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Trees with Equal Restrained Domination and Total Restrained Domination Numbers

For a graph G = (V,E), a set D ⊆ V (G) is a total restrained dominating set if it is a dominating set and both 〈D〉 and 〈V (G)−D〉 do not have isolated vertices. The cardinality of a minimum total restrained dominating set in G is the total restrained domination number. A set D ⊆ V (G) is a restrained dominating set if it is a dominating set and 〈V (G) − D〉 does not contain an isolated vertex. Th...

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Trees with Equal Total Domination and Total Restrained Domination Numbers

For a graph G = (V, E), a set S ⊆ V (G) is a total dominating set if it is dominating and both 〈S〉 has no isolated vertices. The cardinality of a minimum total dominating set in G is the total domination number. A set S ⊆ V (G) is a total restrained dominating set if it is total dominating and 〈V (G) − S〉 has no isolated vertices. The cardinality of a minimum total restrained dominating set in ...

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Total restrained domination numbers of trees

For a given connected graphG= (V ,E), a setDtr ⊆ V (G) is a total restrained dominating set if it is dominating and both 〈Dtr〉 and 〈V (G)−Dtr〉 do not contain isolate vertices. The cardinality of the minimum total restrained dominating set in G is the total restrained domination number and is denoted by tr(G). In this paper we characterize the trees with equal total and total restrained dominati...

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ژورنال

عنوان ژورنال: Discrete Mathematics

سال: 2007

ISSN: 0012-365X

DOI: 10.1016/j.disc.2006.09.014