Displaying similar documents to “Graphs with equal domination and 2-distance domination numbers”

Two Short Proofs on Total Domination

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.

γ-graphs of graphs

Gerd H. Fricke, Sandra M. Hedetniemi, Stephen T. Hedetniemi, Kevin R. Hutson (2011)

Discussiones Mathematicae Graph Theory

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A set S ⊆ V is a dominating set of a graph G = (V,E) if every vertex in V -S is adjacent to at least one vertex in S. The domination number γ(G) of G equals the minimum cardinality of a dominating set S in G; we say that such a set S is a γ-set. In this paper we consider the family of all γ-sets in a graph G and we define the γ-graph G(γ) = (V(γ), E(γ)) of G to be the graph whose vertices V(γ) correspond 1-to-1 with the γ-sets of G, and two γ-sets, say D₁ and D₂, are adjacent in E(γ)...

Total domination subdivision numbers of graphs

Teresa W. Haynes, Michael A. Henning, Lora S. Hopkins (2004)

Discussiones Mathematicae Graph Theory

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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...

Distance 2-Domination in Prisms of Graphs

Ferran Hurtado, Mercè Mora, Eduardo Rivera-Campo, Rita Zuazua (2017)

Discussiones Mathematicae Graph Theory

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A set of vertices D of a graph G is a distance 2-dominating set of G if the distance between each vertex u ∊ (V (G) − D) and D is at most two. Let γ2(G) denote the size of a smallest distance 2-dominating set of G. For any permutation π of the vertex set of G, the prism of G with respect to π is the graph πG obtained from G and a copy G′ of G by joining u ∊ V(G) with v′ ∊ V(G′) if and only if v′ = π(u). If γ2(πG) = γ2(G) for any permutation π of V(G), then G is called a universal γ2-fixer....

Domination and leaf density in graphs

Anders Sune Pedersen (2005)

Discussiones Mathematicae Graph Theory

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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...

Paired-domination

S. Fitzpatrick, B. Hartnell (1998)

Discussiones Mathematicae Graph Theory

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We are interested in dominating sets (of vertices) with the additional property that the vertices in the dominating set can be paired or matched via existing edges in the graph. This could model the situation of guards or police where each has a partner or backup. This paper will focus on those graphs in which the number of matched pairs of a minimum dominating set of this type equals the size of some maximal matching in the graph. In particular, we characterize the leafless graphs of...

The Distance Roman Domination Numbers of Graphs

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...

On the p-domination number of cactus graphs

Mostafa Blidia, Mustapha Chellali, Lutz Volkmann (2005)

Discussiones Mathematicae Graph Theory

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Let p be a positive integer and G = (V,E) a graph. A subset S of V is a p-dominating set if every vertex of V-S is dominated at least p times. The minimum cardinality of a p-dominating set a of G is the p-domination number γₚ(G). It is proved for a cactus graph G that γₚ(G) ⩽ (|V| + |Lₚ(G)| + c(G))/2, for every positive integer p ⩾ 2, where Lₚ(G) is the set of vertices of G of degree at most p-1 and c(G) is the number of odd cycles in G.

On Graphs with Disjoint Dominating and 2-Dominating Sets

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...

The Domination Number of K 3 n

John Georges, Jianwei Lin, David Mauro (2014)

Discussiones Mathematicae Graph Theory

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Let K3n denote the Cartesian product Kn□Kn□Kn, where Kn is the complete graph on n vertices. We show that the domination number of K3n is [...]

On the adjacent eccentric distance sum of graphs

Halina Bielak, Katarzyna Wolska (2015)

Annales UMCS, Mathematica

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In this paper we show bounds for the adjacent eccentric distance sum of graphs in terms of Wiener index, maximum degree and minimum degree. We extend some earlier results of Hua and Yu [Bounds for the Adjacent Eccentric Distance Sum, International Mathematical Forum. Vol. 7 (2O02) no. 26. 1280-1294]. The adjaceni eccentric distance sum index of the graph G is defined as [...] where ε(υ) is the eccentricity of the vertex υ, deg(υ) is the degree of the vertex υ and D(υ) = ∑u∊v(G) d (u,υ)is...

Lower bounds for the domination number

Ermelinda Delaviña, Ryan Pepper, Bill Waller (2010)

Discussiones Mathematicae Graph Theory

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In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matching.