Teresa W. Haynes, Michael A. Henning (2002)
Discussiones Mathematicae Graph Theory
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A set S of vertices of a graph G is a total dominating set if every vertex of V(G) is adjacent to some vertex in S. We provide three equivalent conditions for a tree to have a unique minimum total dominating set and give a constructive characterization of such trees.
Jason T. Hedetniemi (2015)
Discussiones Mathematicae Graph Theory
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Unique minimum vertex dominating sets in the Cartesian product of a graph with a complete graph are considered. We first give properties of such sets when they exist. We then show that when the first factor of the product is a tree, consideration of the tree alone is sufficient to determine if the product has a unique minimum dominating set.
Michael A. Henning, Alister J. Marcon (2016)
Discussiones Mathematicae Graph Theory
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Let G be a graph with no isolated vertex. In this paper, we study a parameter that is squeezed between arguably the two most important domination parameters; namely, the domination number, γ(G), and the total domination number, γt(G). A set S of vertices in a graph G is a semitotal dominating set of G if it is a dominating set of G and every vertex in S is within distance 2 of another vertex of S. The semitotal domination number, γt2(G), is the minimum cardinality of a semitotal dominating...
Mustapha Chellali, Lutz Volkmann (2011)
Discussiones Mathematicae Graph Theory
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Let G = (V(G),E(G)) be a simple graph, and let k be a positive integer. A subset D of V(G) is a k-dominating set if every vertex of V(G) - D is dominated at least k times by D. The k-domination number γₖ(G) is the minimum cardinality of a k-dominating set of G. In [5] Volkmann showed that for every nontrivial tree T, γ₂(T) ≥ γ₁(T)+1 and characterized extremal trees attaining this bound. In this paper we characterize all trees T with γ₂(T) = γ₁(T)+2.
Joanna Raczek (2007)
Discussiones Mathematicae Graph Theory
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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. The cardinality of a minimum restrained dominating set in G is the restrained domination number. We...
Xue-Gang Chen, Wai Chee Shiu, Hong-Yu Chen (2008)
Discussiones Mathematicae Graph Theory
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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...
Michael A. Henning, Douglas F. Rall (2013)
Discussiones Mathematicae Graph Theory
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A DD2-pair of a graph G is a pair (D,D2) of disjoint sets of vertices of G such that D is a dominating set and D2 is a 2-dominating set of G. Although there are infinitely many graphs that do not contain a DD2-pair, we show that every graph with minimum degree at least two has a DD2-pair. We provide a constructive characterization of trees that have a DD2-pair and show that K3,3 is the only connected graph with minimum degree at least three for which D ∪ D2 necessarily contains all vertices...
Dariusz Dereniowski (2009)
Discussiones Mathematicae Graph Theory
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A vertex k-ranking of a simple graph is a coloring of its vertices with k colors in such a way that each path connecting two vertices of the same color contains a vertex with a bigger color. Consider the minimum vertex ranking spanning tree (MVRST) problem where the goal is to find a spanning tree of a given graph G which has a vertex ranking using the minimal number of colors over vertex rankings of all spanning trees of G. K. Miyata et al. proved in [NP-hardness proof and an approximation...
Ladislav Nebeský (1975)
Časopis pro pěstování matematiky
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Bohdan Zelinka (1988)
Czechoslovak Mathematical Journal
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Diana Avella-Alaminos, Magda Dettlaff, Magdalena Lemańska, Rita Zuazua (2015)
Discussiones Mathematicae Graph Theory
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The domination multisubdivision number of a nonempty graph G was defined in [3] as the minimum positive integer k such that there exists an edge which must be subdivided k times to increase the domination number of G. Similarly we define the total domination multisubdivision number msdγt (G) of a graph G and we show that for any connected graph G of order at least two, msdγt (G) ≤ 3. We show that for trees the total domination multisubdi- vision number is equal to the known total domination...
Michael Poschen, Lutz Volkmann (2006)
Discussiones Mathematicae Graph Theory
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Let ir(G) and γ(G) be the irredundance number and domination number of a graph G, respectively. The number of vertices and leaves of a graph G are denoted by n(G) and n₁(G). If T is a tree, then Lemańska [4] presented in 2004 the sharp lower bound γ(T) ≥ (n(T) + 2 - n₁(T))/3. In this paper we prove ir(T) ≥ (n(T) + 2 - n₁(T))/3. for an arbitrary tree T. Since γ(T) ≥ ir(T) is always valid, this inequality is an extension and improvement of...
You Lu, Xinmin Hou, Jun-Ming Xu, Ning Li (2010)
Discussiones Mathematicae Graph Theory
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Let γₜ(G) and γ₂(G) be the total domination number and the 2-domination number of a graph G, respectively. It has been shown that: γₜ(T) ≤ γ₂(T) for any tree T. In this paper, we provide a constructive characterization of those trees with equal total domination number and 2-domination number.