On double domination in graphs

Jochen Harant; Michael A. Henning

Displaying similar documents to “On double domination in graphs”

On locating-domination in graphs

Mustapha Chellali, Malika Mimouni, Peter J. Slater (2010)

Discussiones Mathematicae Graph Theory

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A set D of vertices in a graph G = (V,E) is a locating-dominating set (LDS) if for every two vertices u,v of V-D the sets N(u)∩ D and N(v)∩ D are non-empty and different. The locating-domination number $γ_{L} (G)$ is the minimum cardinality of a LDS of G, and the upper locating-domination number, $Γ_{L} (G)$ is the maximum cardinality of a minimal LDS of G. We present different bounds on $Γ_{L} (G)$ and $γ_{L} (G)$ .

Full domination in graphs

Robert C. Brigham, Gary Chartrand, Ronald D. Dutton, Ping Zhang (2001)

Discussiones Mathematicae Graph Theory

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For each vertex v in a graph G, let there be associated a subgraph $H_{v}$ of G. The vertex v is said to dominate $H_{v}$ as well as dominate each vertex and edge of $H_{v}$ . A set S of vertices of G is called a full dominating set if every vertex of G is dominated by some vertex of S, as is every edge of G. The minimum cardinality of a full dominating set of G is its full domination number $γ_{F H} (G)$ . A full dominating set of G of cardinality $γ_{F H} (G)$ is called a $γ_{F H}$ -set of G. We study three types of full domination in...

On the adjacent eccentric distance sum of graphs

Halina Bielak, Katarzyna Wolska (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – 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 (2002) no. 26, 1289–1294]. The adjacent eccentric distance sum index of the graph $G$ is defined as $ξ^{s v} (G) = \sum_{v \in V (G)} \frac{ε (v) D (v)}{d e g (v)},$ where $ε (v)$ is the eccentricity of the vertex $v$ , $d e g (v)$ is the degree of the vertex $v$ and $D (v) = \sum_{u \in V (G)} d (u, v)$ is the sum of all distances from...

Roman bondage in graphs

Nader Jafari Rad, Lutz Volkmann (2011)

Discussiones Mathematicae Graph Theory

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A Roman dominating function on a graph G is a function f:V(G) → 0,1,2 satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. The weight of a Roman dominating function is the value $f (V (G)) = \sum_{u \in V (G)} f (u)$ . The Roman domination number, $γ_{R} (G)$ , of G is the minimum weight of a Roman dominating function on G. In this paper, we define the Roman bondage $b_{R} (G)$ of a graph G with maximum degree at least two to be the minimum cardinality of all sets E’ ⊆ E(G)...

Some remarks on α-domination

Franz Dahme, Dieter Rautenbach, Lutz Volkmann (2004)

Discussiones Mathematicae Graph Theory

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Let α ∈ (0,1) and let $G = (V_{G}, E_{G}$ ) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set $D \subseteq V_{G}$ is called an α-dominating set of G, if $| N_{G} (u) \cap D | \geq α d_{G} (u)$ for all $u \in V_{G} ∖ D$ . We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.

Domination and independence subdivision numbers of graphs

Teresa W. Haynes, Sandra M. Hedetniemi, Stephen T. Hedetniemi (2000)

Discussiones Mathematicae Graph Theory

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The domination subdivision number $s d_{γ} (G)$ of a graph is the minimum number of edges that must be subdivided (where an edge can be subdivided at most once) in order to increase the domination number. Arumugam showed that this number is at most three for any tree, and conjectured that the upper bound of three holds for any graph. Although we do not prove this interesting conjecture, we give an upper bound for the domination subdivision number for any graph G in terms of the minimum degrees of...

Secure domination and secure total domination in graphs

William F. Klostermeyer, Christina M. Mynhardt (2008)

Discussiones Mathematicae Graph Theory

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A secure (total) dominating set of a graph G = (V,E) is a (total) dominating set X ⊆ V with the property that for each u ∈ V-X, there exists x ∈ X adjacent to u such that $(X - x) \cup u$ is (total) dominating. The smallest cardinality of a secure (total) dominating set is the secure (total) domination number $γ_{s} (G) (γ_{s t} (G))$ . We characterize graphs with equal total and secure total domination numbers. We show that if G has minimum degree at least two, then $γ_{s t} (G) \leq γ_{s} (G)$ . We also show that $γ_{s t} (G)$ is at most twice the clique covering...

Stronger bounds for generalized degrees and Menger path systems

R.J. Faudree, Zs. Tuza (1995)

Discussiones Mathematicae Graph Theory

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For positive integers d and m, let $P_{d, m} (G)$ denote the property that between each pair of vertices of the graph G, there are m internally vertex disjoint paths of length at most d. For a positive integer t a graph G satisfies the minimum generalized degree condition δₜ(G) ≥ s if the cardinality of the union of the neighborhoods of each set of t vertices of G is at least s. Generalized degree conditions that ensure that $P_{d, m} (G)$ is satisfied have been investigated. In particular, it has been shown,...

The Turán number of the graph $3 P_{4}$

Halina Bielak, Sebastian Kieliszek (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $e x (n, G)$ denote the maximum number of edges in a graph on $n$ vertices which does not contain $G$ as a subgraph. Let $P_{i}$ denote a path consisting of $i$ vertices and let $m P_{i}$ denote $m$ disjoint copies of $P_{i}$ . In this paper we count $e x (n, 3 P_{4})$ .

A note on the independent domination number versus the domination number in bipartite graphs

Shaohui Wang, Bing Wei (2017)

Czechoslovak Mathematical Journal

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Let $γ (G)$ and $i (G)$ be the domination number and the independent domination number of $G$ , respectively. Rad and Volkmann posted a conjecture that $i (G) / γ (G) \leq Δ (G) / 2$ for any graph $G$ , where $Δ (G)$ is its maximum degree (see N. J. Rad, L. Volkmann (2013)). In this work, we verify the conjecture for bipartite graphs. Several graph classes attaining the extremal bound and graphs containing odd cycles with the ratio larger than $Δ (G) / 2$ are provided as well.

Distance independence in graphs

J. Louis Sewell, Peter J. Slater (2011)

Discussiones Mathematicae Graph Theory

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For a set D of positive integers, we define a vertex set S ⊆ V(G) to be D-independent if u, v ∈ S implies the distance d(u,v) ∉ D. The D-independence number $β_{D} (G)$ is the maximum cardinality of a D-independent set. In particular, the independence number $β (G) = β_{1} (G)$ . Along with general results we consider, in particular, the odd-independence number $β_{O D D} (G)$ where ODD = 1,3,5,....

Some properties of generalized distance eigenvalues of graphs

Yuzheng Ma, Yan Ling Shao (2024)

Czechoslovak Mathematical Journal

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Let $G$ be a simple connected graph with vertex set $V (G) = {v_{1}, v_{2}, \dots, v_{n}}$ and edge set $E (G)$ , and let $d_{v_{i}}$ be the degree of the vertex $v_{i}$ . Let $D (G)$ be the distance matrix and let $T_{r} (G)$ be the diagonal matrix of the vertex transmissions of $G$ . The generalized distance matrix of $G$ is defined as $D_{α} (G) = α T_{r} (G) + (1 - α) D (G)$ , where $0 \leq α \leq 1$ . Let $λ_{1} (D_{α} (G)) \geq λ_{2} (D_{α} (G)) \geq ... \geq λ_{n} (D_{α} (G))$ be the generalized distance eigenvalues of $G$ , and let $k$ be an integer with $1 \leq k \leq n$ . We denote by $S_{k} (D_{α} (G)) = λ_{1} (D_{α} (G)) + λ_{2} (D_{α} (G)) + ... + λ_{k} (D_{α} (G))$ the sum of the $k$ largest generalized distance eigenvalues. The generalized distance spread of a graph $G$ is defined as $D_{α} S (G) = λ_{1} (D_{α} (G)) - λ_{n} (D_{α} (G))$ ....

On path-quasar Ramsey numbers

Binlong Li, Bo Ning (2014)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let $G_{1}$ and $G_{2}$ be two given graphs. The Ramsey number $R (G_{1}, G_{2})$ is the least integer $r$ such that for every graph $G$ on $r$ vertices, either $G$ contains a $G_{1}$ or $\bar{G}$ contains a $G_{2}$ . Parsons gave a recursive formula to determine the values of $R (P_{n}, K_{1, m})$ , where $P_{n}$ is a path on $n$ vertices and $K_{1, m}$ is a star on $m + 1$ vertices. In this note, we study the Ramsey numbers $R (P_{n}, K_{1} \lor F_{m})$ , where $F_{m}$ is a linear forest on $m$ vertices. We determine the exact values of $R (P_{n}, K_{1} \lor F_{m})$ for the cases $m \leq n$ and $m \geq 2 n$ , and for the case that $F_{m}$ has no odd component. Moreover, we...

Edit distance measure for graphs

Tomasz Dzido, Krzysztof Krzywdziński (2015)

Czechoslovak Mathematical Journal

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In this paper, we investigate a measure of similarity of graphs similar to the Ramsey number. We present values and bounds for $g (n, l)$ , the biggest number $k$ guaranteeing that there exist $l$ graphs on $n$ vertices, each two having edit distance at least $k$ . By edit distance of two graphs $G$ , $F$ we mean the number of edges needed to be added to or deleted from graph $G$ to obtain graph $F$ . This new extremal number $g (n, l)$ is closely linked to the edit distance of graphs. Using probabilistic methods we show...

Several quantitative characterizations of some specific groups

A. Mohammadzadeh, Ali Reza Moghaddamfar (2017)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a finite group and let $π (G) = {p_{1}, p_{2}, ..., p_{k}}$ be the set of prime divisors of $| G |$ for which $p_{1} < p_{2} < \dots < p_{k}$ . The Gruenberg-Kegel graph of $G$ , denoted $GK (G)$ , is defined as follows: its vertex set is $π (G)$ and two different vertices $p_{i}$ and $p_{j}$ are adjacent by an edge if and only if $G$ contains an element of order $p_{i} p_{j}$ . The degree of a vertex $p_{i}$ in $GK (G)$ is denoted by $d_{G} (p_{i})$ and the $k$ -tuple $D (G) = (d_{G} (p_{1}), d_{G} (p_{2}), ..., d_{G} (p_{k}))$ is said to be the degree pattern of $G$ . Moreover, if $ω \subseteq π (G)$ is the vertex set of a connected component of $GK (G)$ , then the largest $ω$ -number which divides $| G |$ , is...

On the total k-domination number of graphs

Adel P. Kazemi (2012)

Discussiones Mathematicae Graph Theory

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Let k be a positive integer and let G = (V,E) be a simple graph. The k-tuple domination number $γ_{\times k} (G)$ of G is the minimum cardinality of a k-tuple dominating set S, a set that for every vertex v ∈ V, $| N_{G} [v] \cap S | \geq k$ . Also the total k-domination number $γ_{\times k, t} (G)$ of G is the minimum cardinality of a total k -dominating set S, a set that for every vertex v ∈ V, $| N_{G} (v) \cap S | \geq k$ . The k-transversal number τₖ(H) of a hypergraph H is the minimum size of a subset S ⊆ V(H) such that |S ∩e | ≥ k for every edge e ∈ E(H). We know that for...

2-factors in claw-free graphs with locally disconnected vertices

Mingqiang An, Liming Xiong, Runli Tian (2015)

Czechoslovak Mathematical Journal

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An edge of $G$ is singular if it does not lie on any triangle of $G$ ; otherwise, it is non-singular. A vertex $u$ of a graph $G$ is called locally connected if the induced subgraph $G [N (u)]$ by its neighborhood is connected; otherwise, it is called locally disconnected. In this paper, we prove that if a connected claw-free graph $G$ of order at least three satisfies the following two conditions: (i) for each locally disconnected vertex $v$ of degree at least $3$ in $G,$ there is a nonnegative integer $s$ such...