Distance in graphs
Roger C. Entringer, Douglas E. Jackson, D. A. Snyder (1976)
Czechoslovak Mathematical Journal
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Roger C. Entringer, Douglas E. Jackson, D. A. Snyder (1976)
Czechoslovak Mathematical Journal
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Allan Bickle (2013)
Discussiones Mathematicae Graph Theory
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A set of vertices of a graph G is a total dominating set if each vertex of G is adjacent to a vertex in the set. The total domination number of a graph Υt (G) is the minimum size of a total dominating set. We provide a short proof of the result that Υt (G) ≤ 2/3n for connected graphs with n ≥ 3 and a short characterization of the extremal graphs.
K. CH. Das, I. Gutman, D. Vukičević (2011)
Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences mathématiques
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Hopkins, Glenn, Staton, William (1989)
International Journal of Mathematics and Mathematical Sciences
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Bohdan Zelinka (1976)
Mathematica Slovaca
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Joanna Cyman, Magdalena Lemańska, Joanna Raczek (2006)
Open Mathematics
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For a given connected graph G = (V, E), a set is a doubly connected dominating set if it is dominating and both 〈D〉 and 〈V (G)-D〉 are connected. The cardinality of the minimum doubly connected dominating set in G is the doubly connected domination number. We investigate several properties of doubly connected dominating sets and give some bounds on the doubly connected domination number.
A.P. Santhakumaran, P. Titus (2012)
Discussiones Mathematicae Graph Theory
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For a connected graph G of order p ≥ 2 and a vertex x of G, a set S ⊆ V(G) is an x-monophonic set of G if each vertex v ∈ V(G) lies on an x -y monophonic path for some element y in S. The minimum cardinality of an x-monophonic set of G is defined as the x-monophonic number of G, denoted by mₓ(G). An x-monophonic set of cardinality mₓ(G) is called a mₓ-set of G. We determine bounds for it and characterize graphs which realize these bounds. A connected graph of order p with vertex monophonic...
Wenjing Li, Xueliang Li, Jingshu Zhang (2018)
Discussiones Mathematicae Graph Theory
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A path in a vertex-colored graph is called vertex-rainbow if its internal vertices have pairwise distinct colors. A vertex-colored graph G is rainbow vertex-connected if for any two distinct vertices of G, there is a vertex-rainbow path connecting them. For a connected graph G, the rainbow vertex-connection number of G, denoted by rvc(G), is defined as the minimum number of colors that are required to make G rainbow vertex-connected. In this paper, we find all the families ℱ of connected...
Hamideh Aram, Sepideh Norouzian, Seyed Mahmoud Sheikholeslami (2013)
Discussiones Mathematicae Graph Theory
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Let k be a positive integer, and let G be a simple graph with vertex set V (G). A k-distance Roman dominating function on G is a labeling f : V (G) → {0, 1, 2} such that for every vertex with label 0, there is a vertex with label 2 at distance at most k from each other. The weight of a k-distance Roman dominating function f is the value w(f) =∑v∈V f(v). The k-distance Roman domination number of a graph G, denoted by γkR (D), equals the minimum weight of a k-distance Roman dominating...