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1
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Total domination subdivision numbers of graphs

100%
Haynes T., Henning M., Hopkins L.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
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nr 3
457-467
EN
A set S of vertices in a graph G = (V,E) is a total dominating set of G if every vertex of V is adjacent to a vertex in S. The total domination number of G is the minimum cardinality of a total dominating set of G. The total domination subdivision number of G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. First we establish bounds on the total domination subdivision number for some families of graphs. Then we show that the total domination subdivision number of a graph can be arbitrarily large.
2
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Total domination of Cartesian products of graphs

80%
Hou X.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 1
175-178
EN
Let γₜ(G) and $γ_{pr}(G)$ denote the total domination and the paired domination numbers of graph G, respectively, and let G □ H denote the Cartesian product of graphs G and H. In this paper, we show that γₜ(G)γₜ(H) ≤ 5γₜ(G □ H), which improves the known result γₜ(G)γₜ(H) ≤ 6γₜ(G □ H) given by Henning and Rall.
3
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Trees with equal total domination and total restrained domination numbers

80%
Chen X.-G., Shiu W., Chen H.-Y.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 1
59-66
EN
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 G is the total restrained domination number. We characterize all trees for which total domination and total restrained domination numbers are the same.
4
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On the total restrained domination number of direct products of graphs

80%
Shiu W., Chen H.-Y., Chen X.-G., Sun P.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
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nr 4
629-641
EN
Let G = (V,E) be a graph. A total restrained dominating set is a set S ⊆ V where every vertex in V∖S is adjacent to a vertex in S as well as to another vertex in V∖S, and every vertex in S is adjacent to another vertex in S. The total restrained domination number of G, denoted by $γ_r^t(G)$, is the smallest cardinality of a total restrained dominating set of G. We determine lower and upper bounds on the total restrained domination number of the direct product of two graphs. Also, we show that these bounds are sharp by presenting some infinite families of graphs that attain these bounds.
5
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Matchings and total domination subdivision number in graphs with few induced 4-cycles

80%
Favaron O., Karami H., Khoeilar R., Sheikholeslami S.
Discussiones Mathematicae Graph Theory
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2010
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tom 30
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nr 4
611-618
EN
A set S of vertices of a graph G = (V,E) without isolated vertex is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. The total domination number γₜ(G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number $sd_{γₜ(G)}$ is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the total domination number. Favaron, Karami, Khoeilar and Sheikholeslami (Journal of Combinatorial Optimization, to appear) conjectured that: For any connected graph G of order n ≥ 3, $sd_{γₜ(G)} ≤ γₜ(G)+1$. In this paper we use matchings to prove this conjecture for graphs with at most three induced 4-cycles through each vertex.
6
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A Gallai-type equality for the total domination number of a graph

80%
Zhou S.
Discussiones Mathematicae Graph Theory
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2004
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tom 24
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nr 3
539-543
EN
We prove the following Gallai-type equality γₜ(G) + εₜ(G) = p for any graph G with no isolated vertex, where p is the number of vertices of G, γₜ(G) is the total domination number of G, and εₜ(G) is the maximum integer s such that there exists a spanning forest F with s the number of pendant edges of F minus the number of star components of F.
7
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On the (2,2)-domination number of trees

80%
Lu Y., Hou X., Xu J.-M.
Discussiones Mathematicae Graph Theory
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2010
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tom 30
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nr 2
185-199
EN
Let γ(G) and $γ_{2,2}(G)$ denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that $(2(γ(T)+1))/3 ≤ γ_{2,2}(T) ≤ 2γ(T)$. Moreover, we characterize all the trees achieving the equalities.
8
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Domination and leaf density in graphs

70%
Pedersen A.
Discussiones Mathematicae Graph Theory
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2005
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tom 25
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nr 3
251-259
EN
The domination number γ(G) of a graph G is the minimum cardinality of a subset D of V(G) with the property that each vertex of V(G)-D is adjacent to at least one vertex of D. For a graph G with n vertices we define ε(G) to be the number of leaves in G minus the number of stems in G, and we define the leaf density ζ(G) to equal ε(G)/n. We prove that for any graph G with no isolated vertex, γ(G) ≤ n(1- ζ(G))/2 and we characterize the extremal graphs for this bound. Similar results are obtained for the total domination number and the partition domination number.
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