Displaying similar documents to “Trees with unique minimum total dominating sets”

Vertices Contained In All Or In No Minimum Semitotal Dominating Set Of A Tree

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...

Trees with equal 2-domination and 2-independence numbers

Mustapha Chellali, Nacéra Meddah (2012)

Discussiones Mathematicae Graph Theory

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Let G = (V,E) be a graph. A subset S of V is a 2-dominating set if every vertex of V-S is dominated at least 2 times, and S is a 2-independent set of G if every vertex of S has at most one neighbor in S. The minimum cardinality of a 2-dominating set a of G is the 2-domination number γ₂(G) and the maximum cardinality of a 2-independent set of G is the 2-independence number β₂(G). Fink and Jacobson proved that γ₂(G) ≤ β₂(G) for every graph G. In this paper we provide a constructive characterization...

Minimum vertex ranking spanning tree problem for chordal and proper interval graphs

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...

Trees with equal total domination and total restrained domination numbers

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...

Spanning Trees whose Stems have a Bounded Number of Branch Vertices

Zheng Yan (2016)

Discussiones Mathematicae Graph Theory

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Let T be a tree, a vertex of degree one and a vertex of degree at least three is called a leaf and a branch vertex, respectively. The set of leaves of T is denoted by Leaf(T). The subtree T − Leaf(T) of T is called the stem of T and denoted by Stem(T). In this paper, we give two sufficient conditions for a connected graph to have a spanning tree whose stem has a bounded number of branch vertices, and these conditions are best possible.

Trees with equal restrained domination and total restrained domination numbers

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...

Caterpillars

Bohdan Zelinka (1977)

Časopis pro pěstování matematiky

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Global alliances and independence in trees

Mustapha Chellali, Teresa W. Haynes (2007)

Discussiones Mathematicae Graph Theory

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A global defensive (respectively, offensive) alliance in a graph G = (V,E) is a set of vertices S ⊆ V with the properties that every vertex in V-S has at least one neighbor in S, and for each vertex v in S (respectively, in V-S) at least half the vertices from the closed neighborhood of v are in S. These alliances are called strong if a strict majority of vertices from the closed neighborhood of v must be in S. For each kind of alliance, the associated parameter is the minimum cardinality...