On the (2,2)-domination number of trees

You Lu; Xinmin Hou; Jun-Ming Xu

Displaying similar documents to “On the (2,2)-domination number of trees”

On locating and differentiating-total domination in trees

Mustapha Chellali (2008)

Discussiones Mathematicae Graph Theory

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A total dominating set of a graph G = (V,E) with no isolated vertex is a set S ⊆ V such that every vertex is adjacent to a vertex in S. A total dominating set S of a graph G is a locating-total dominating set if for every pair of distinct vertices u and v in V-S, N(u)∩S ≠ N(v)∩S, and S is a differentiating-total dominating set if for every pair of distinct vertices u and v in V, N[u]∩S ≠ N[v] ∩S. Let $γ ₜ^{L} (G)$ and $γ ₜ^{D} (G)$ be the minimum cardinality of a locating-total dominating set and a differentiating-total...

A note on the cubical dimension of new classes of binary trees

Kamal Kabyl, Abdelhafid Berrachedi, Éric Sopena (2015)

Czechoslovak Mathematical Journal

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The cubical dimension of a graph $G$ is the smallest dimension of a hypercube into which $G$ is embeddable as a subgraph. The conjecture of Havel (1984) claims that the cubical dimension of every balanced binary tree with $2^{n}$ vertices, $n \geq 1$ , is $n$ . The 2-rooted complete binary tree of depth $n$ is obtained from two copies of the complete binary tree of depth $n$ by adding an edge linking their respective roots. In this paper, we determine the cubical dimension of trees obtained by subdividing twice...

Closure for spanning trees and distant area

Jun Fujisawa, Akira Saito, Ingo Schiermeyer (2011)

Discussiones Mathematicae Graph Theory

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A k-ended tree is a tree with at most k endvertices. Broersma and Tuinstra [3] have proved that for k ≥ 2 and for a pair of nonadjacent vertices u, v in a graph G of order n with $d e g_{G} u + d e g_{G} v \geq n - 1$ , G has a spanning k-ended tree if and only if G+uv has a spanning k-ended tree. The distant area for u and v is the subgraph induced by the set of vertices that are not adjacent with u or v. We investigate the relationship between the condition on $d e g_{G} u + d e g_{G} v$ and the structure of the distant area for u and v. We prove...

Turán's problem and Ramsey numbers for trees

Zhi-Hong Sun, Lin-Lin Wang, Yi-Li Wu (2015)

Colloquium Mathematicae

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Let T¹ₙ = (V,E₁) and T²ₙ = (V,E₂) be the trees on n vertices with $V = v ₀, v ₁, . . ., v_{n - 1}$ , $E ₁ = v ₀ v ₁, . . ., v ₀ v_{n - 3}, v_{n - 4} v_{n - 2}, v_{n - 3} v_{n - 1}$ and $E ₂ = v ₀ v ₁, . . ., v ₀ v_{n - 3}, v_{n - 3} v_{n - 2}, v_{n - 3} v_{n - 1}$ . For p ≥ n ≥ 5 we obtain explicit formulas for ex(p;T¹ₙ) and ex(p;T²ₙ), where ex(p;L) denotes the maximal number of edges in a graph of order p not containing L as a subgraph. Let r(G₁,G₂) be the Ramsey number of the two graphs G₁ and G₂. We also obtain some explicit formulas for $r (T ₘ, T ₙ^{i})$ , where i ∈ 1,2 and Tₘ is a tree on m vertices with Δ(Tₘ) ≤ m - 3.

The tree property at both $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$

Laura Fontanella, Sy David Friedman (2015)

Fundamenta Mathematicae

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We force from large cardinals a model of ZFC in which $ℵ_{ω + 1}$ and $ℵ_{ω + 2}$ both have the tree property. We also prove that if we strengthen the large cardinal assumptions, then in the final model $ℵ_{ω + 2}$ even satisfies the super tree property.

Quasi-tree graphs with the minimal Sombor indices

Yibo Li, Huiqing Liu, Ruiting Zhang (2022)

Czechoslovak Mathematical Journal

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The Sombor index $S O (G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d_{G}^{2} (u) + d_{G}^{2} (v)}$ of all edges $u v$ of $G$ , where $d_{G} (u)$ denotes the degree of the vertex $u$ in $G$ . A connected graph $G = (V, E)$ is called a quasi-tree if there exists $u \in V (G)$ such that $G - u$ is a tree. Denote $𝒬 (n, k) = {G : G$ is a quasi-tree graph of order $n$ with $G - u$ being a tree and $d_{G} (u) = k}$ . We determined the minimum and the second minimum Sombor indices of all quasi-trees in $𝒬 (n, k)$ . Furthermore, we characterized the corresponding extremal graphs, respectively.

On a characterization of $k$ -trees

De-Yan Zeng, Jian Hua Yin (2015)

Czechoslovak Mathematical Journal

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A graph $G$ is a $k$ -tree if either $G$ is the complete graph on $k + 1$ vertices, or $G$ has a vertex $v$ whose neighborhood is a clique of order $k$ and the graph obtained by removing $v$ from $G$ is also a $k$ -tree. Clearly, a $k$ -tree has at least $k + 1$ vertices, and $G$ is a 1-tree (usual tree) if and only if it is a $1$ -connected graph and has no $K_{3}$ -minor. In this paper, motivated by some properties of 2-trees, we obtain a characterization of $k$ -trees as follows: if $G$ is a graph with at least $k + 1$ vertices, then $G$ is...

A lower bound for the 3-pendant tree-connectivity of lexicographic product graphs

Yaping Mao, Christopher Melekian, Eddie Cheng (2023)

Czechoslovak Mathematical Journal

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For a connected graph $G = (V, E)$ and a set $S \subseteq V (G)$ with at least two vertices, an $S$ -Steiner tree is a subgraph $T = (V^{'}, E^{'})$ of $G$ that is a tree with $S \subseteq V^{'}$ . If the degree of each vertex of $S$ in $T$ is equal to 1, then $T$ is called a pendant $S$ -Steiner tree. Two $S$ -Steiner trees are if they share no vertices other than $S$ and have no edges in common. For $S \subseteq V (G)$ and $| S | \geq 2$ , the pendant tree-connectivity $τ_{G} (S)$ is the maximum number of internally disjoint pendant $S$ -Steiner trees in $G$ , and for $k \geq 2$ , the $k$ -pendant tree-connectivity $τ_{k} (G)$ is the...

Extremal trees and molecular trees with respect to the Sombor-index-like graph invariants ${𝒮𝒪}_{5}$ and ${𝒮𝒪}_{6}$

Wei Gao (2024)

Czechoslovak Mathematical Journal

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I. Gutman (2022) constructed six new graph invariants based on geometric parameters, and named them Sombor-index-like graph invariants, denoted by ${𝒮𝒪}_{1}, {𝒮𝒪}_{2}, \dots, {𝒮𝒪}_{6}$ . Z. Tang, H. Deng (2022) and Z. Tang, Q. Li, H. Deng (2023) investigated the chemical applicability and extremal values of these Sombor-index-like graph invariants, and raised some open problems, see Z. Tang, Q. Li, H. Deng (2023). We consider the first open problem formulated at the end of Z. Tang, Q. Li, H. Deng (2023). We obtain the extremal...

The instability of nonseparable complete Erdős spaces and representations in ℝ-trees

Jan J. Dijkstra, Kirsten I. S. Valkenburg (2010)

Fundamenta Mathematicae

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One way to generalize complete Erdős space $_{c}$ is to consider uncountable products of zero-dimensional $G_{δ}$ -subsets of the real line, intersected with an appropriate Banach space. The resulting (nonseparable) complete Erdős spaces can be fully classified by only two cardinal invariants, as done in an earlier paper of the authors together with J. van Mill. As we think this is the correct way to generalize the concept of complete Erdős space to a nonseparable setting, natural questions arise...

The relation between the number of leaves of a tree and its diameter

Pu Qiao, Xingzhi Zhan (2022)

Czechoslovak Mathematical Journal

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Let $L (n, d)$ denote the minimum possible number of leaves in a tree of order $n$ and diameter $d .$ Lesniak (1975) gave the lower bound $B (n, d) = ⌈ 2 (n - 1) / d ⌉$ for $L (n, d) .$ When $d$ is even, $B (n, d) = L (n, d) .$ But when $d$ is odd, $B (n, d)$ is smaller than $L (n, d)$ in general. For example, $B (21, 3) = 14$ while $L (21, 3) = 19 .$ In this note, we determine $L (n, d)$ using new ideas. We also consider the converse problem and determine the minimum possible diameter of a tree with given order and number of leaves.

Trees and the dynamics of polynomials

Laura G. DeMarco, Curtis T. McMullen (2008)

Annales scientifiques de l'École Normale Supérieure

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In this paper we study branched coverings of metrized, simplicial trees $F : T \to T$ which arise from polynomial maps $f : ℂ \to ℂ$ with disconnected Julia sets. We show that the collection of all such trees, up to scale, forms a contractible space $ℙ T_{D}$ compactifying the moduli space of polynomials of degree $D$ ; that $F$ records the asymptotic behavior of the multipliers of $f$ ; and that any meromorphic family of polynomials over $Δ^{*}$ can be completed by a unique tree at its central fiber. In the cubic case we give a...

Shadow trees of Mandelbrot sets

Virpi Kauko (2003)

Fundamenta Mathematicae

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The topology and combinatorial structure of the Mandelbrot set $ℳ^{d}$ (of degree d ≥ 2) can be studied using symbolic dynamics. Each parameter is mapped to a kneading sequence, or equivalently, an internal address; but not every such sequence is realized by a parameter in $ℳ^{d}$ . Thus the abstract Mandelbrot set is a subspace of a larger, partially ordered symbol space, $Λ^{d}$ . In this paper we find an algorithm to construct “visible trees” from symbolic sequences which works whether or not the sequence...

On operators which factor through $l_{p}$ or c₀

Bentuo Zheng (2006)

Studia Mathematica

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Let 1 < p < ∞. Let X be a subspace of a space Z with a shrinking F.D.D. (Eₙ) which satisfies a block lower-p estimate. Then any bounded linear operator T from X which satisfies an upper-(C,p)-tree estimate factors through a subspace of ${(\sum F ₙ)}_{l_{p}}$ , where (Fₙ) is a blocking of (Eₙ). In particular, we prove that an operator from $L_{p}$ (2 < p < ∞) satisfies an upper-(C,p)-tree estimate if and only if it factors through $l_{p}$ . This gives an answer to a question of W. B. Johnson. We also prove...