Upper bounds for the domination numbers of toroidal queens graphs

Christina M. Mynhardt

Displaying similar documents to “Upper bounds for the domination numbers of toroidal queens graphs”

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.

On 𝓕-independence in graphs

Frank Göring, Jochen Harant, Dieter Rautenbach, Ingo Schiermeyer (2009)

Discussiones Mathematicae Graph Theory

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Let be a set of graphs and for a graph G let $α_{} (G)$ and $α *_{} (G)$ denote the maximum order of an induced subgraph of G which does not contain a graph in as a subgraph and which does not contain a graph in as an induced subgraph, respectively. Lower bounds on $α_{} (G)$ and $α *_{} (G)$ are presented.

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 $γ_{L} (G)$ .

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...

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...

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.

On the diameter of the intersection graph of a finite simple group

Xuanlong Ma (2016)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group. The intersection graph $Δ_{G}$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of $G$ , and two distinct vertices $X$ and $Y$ are adjacent if $X \cap Y \neq 1$ , where $1$ denotes the trivial subgroup of order $1$ . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters...

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)...

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...

A spectral bound for graph irregularity

Felix Goldberg (2015)

Czechoslovak Mathematical Journal

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The imbalance of an edge $e = {u, v}$ in a graph is defined as $i (e) = | d (u) - d (v) |$ , where $d (\cdot)$ is the vertex degree. The irregularity $I (G)$ of $G$ is then defined as the sum of imbalances over all edges of $G$ . This concept was introduced by Albertson who proved that $I (G) \leq 4 n^{3} / 27$ (where $n = | V (G) |$ ) and obtained stronger bounds for bipartite and triangle-free graphs. Since then a number of additional bounds were given by various authors. In this paper we prove a new upper bound, which improves a bound found by Zhou and Luo in 2008. Our bound involves...

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...

Remarks on $D$ -integral complete multipartite graphs

Pavel Híc, Milan Pokorný (2016)

Czechoslovak Mathematical Journal

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A graph is called distance integral (or $D$ -integral) if all eigenvalues of its distance matrix are integers. In their study of $D$ -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on $D$ -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs $K_{p_{1}, p_{2}, p_{3}}$ with $p_{1} < p_{2} < p_{3}$ , and $K_{p_{1}, p_{2}, p_{3}, p_{4}}$ with $p_{1} < p_{2} < p_{3} < p_{4}$ , as well as the infinite classes of distance integral...

On double domination in graphs

Jochen Harant, Michael A. Henning (2005)

Discussiones Mathematicae Graph Theory

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In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number $γ_{\times 2} (G)$ . A function f(p) is defined, and it is shown that $γ_{\times 2} (G) = m i n f (p)$ , where the minimum is taken over the n-dimensional cube $C ⁿ = p = (p ₁, . . ., p ₙ) | p_{i} \in I R, 0 \leq p_{i} \leq 1, i = 1, . . ., n$ . Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then $γ_{\times 2} (G) \leq ((l n (1 + d) + l n δ + 1) / δ) n$ .

On the bounds of Laplacian eigenvalues of $k$ -connected graphs

Xiaodan Chen, Yaoping Hou (2015)

Czechoslovak Mathematical Journal

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Let $μ_{n - 1} (G)$ be the algebraic connectivity, and let $μ_{1} (G)$ be the Laplacian spectral radius of a $k$ -connected graph $G$ with $n$ vertices and $m$ edges. In this paper, we prove that $μ_{n - 1} (G) \geq \frac{2 n k^{2}}{(n (n - 1) - 2 m) (n + k - 2) + 2 k^{2}},$ with equality if and only if $G$ is the complete graph $K_{n}$ or $K_{n} - e$ . Moreover, if $G$ is non-regular, then $μ_{1} (G) < 2 Δ - \frac{2 (n Δ - 2 m) k^{2}}{2 (n Δ - 2 m) (n^{2} - 2 n + 2 k) + n k^{2}},$ where $Δ$ stands for the maximum degree of $G$ . Remark that in some cases, these two inequalities improve some previously known results.

Paired domination in prisms of graphs

Christina M. Mynhardt, Mark Schurch (2011)

Discussiones Mathematicae Graph Theory

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The paired domination number $γ_{p r} (G)$ of a graph G is the smallest cardinality of a dominating set S of G such that ⟨S⟩ has a perfect matching. The generalized prisms πG of G are the graphs obtained by joining the vertices of two disjoint copies of G by |V(G)| independent edges. We provide characterizations of the following three classes of graphs: $γ_{p r} (π G) = 2 γ_{p r} (G)$ for all πG; $γ_{p r} (K ₂ ☐ G) = 2 γ_{p r} (G)$ ; $γ_{p r} (K ₂ ☐ G) = γ_{p r} (G)$ .

Upper oriented chromatic number of undirected graphs and oriented colorings of product graphs

Éric Sopena (2012)

Discussiones Mathematicae Graph Theory

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The oriented chromatic number of an oriented graph $^{\to} G$ is the minimum order of an oriented graph $^{\to} H$ such that $^{\to} G$ admits a homomorphism to $^{\to} H$ . The oriented chromatic number of an undirected graph G is then the greatest oriented chromatic number of its orientations. In this paper, we introduce the new notion of the upper oriented chromatic number of an undirected graph G, defined as the minimum order of an oriented graph $^{\to} U$ such that every orientation $^{\to} G$ of G admits a homomorphism to $^{\to} U$ . We give...

Intrinsic linking and knotting are arbitrarily complex

Erica Flapan, Blake Mellor, Ramin Naimi (2008)

Fundamenta Mathematicae

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We show that, given any n and α, any embedding of any sufficiently large complete graph in ℝ³ contains an oriented link with components Q₁, ..., Qₙ such that for every i ≠ j, $| l k (Q_{i}, Q_{j}) | \geq α$ and $| a ₂ (Q_{i}) | \geq α$ , where $a ₂ (Q_{i})$ denotes the second coefficient of the Conway polynomial of $Q_{i}$ .