Closure for spanning trees and distant area

Jun Fujisawa; Akira Saito; Ingo Schiermeyer

Displaying similar documents to “Closure for spanning trees and distant area”

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

You Lu, Xinmin Hou, Jun-Ming Xu (2010)

Discussiones Mathematicae Graph Theory

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Let γ(G) and $γ_{2, 2} (G)$ denote the domination number and (2,2)-domination number of a graph G, respectively. In this paper, for any nontrivial tree T, we show that $(2 (γ (T) + 1)) / 3 \leq γ_{2, 2} (T) \leq 2 γ (T)$ . Moreover, we characterize all the trees achieving the equalities.

Tree pattern matching from regular tree expressions

Ahlem Belabbaci, Hadda Cherroun, Loek Cleophas, Djelloul Ziadi (2018)

Kybernetika

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In this work we deal with tree pattern matching over ranked trees, where the pattern set to be matched against is defined by a regular tree expression. We present a new method that uses a tree automaton constructed inductively from a regular tree expression. First we construct a special tree automaton for the regular tree expression of the pattern $E$ , which is somehow a generalization of Thompson automaton for strings. Then we run the constructed automaton on the subject tree $t$ . The pattern...

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...

Signpost systems and spanning trees of graphs

Ladislav Nebeský (2006)

Czechoslovak Mathematical Journal

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By a ternary system we mean an ordered pair $(W, R)$ , where $W$ is a finite nonempty set and $R \subseteq W \times W \times W$ . By a signpost system we mean a ternary system $(W, R)$ satisfying the following conditions for all $x, y, z \in W$ : if $(x, y, z) \in R$ , then $(y, x, x) \in R$ and $(y, x, z) \notin R$ ; if $x \neq y$ , then there exists $t \in W$ such that $(x, t, y) \in R$ . In this paper, a signpost system is used as a common description of a connected graph and a spanning tree of the graph. By a ct-pair we mean an ordered pair $(G, T)$ , where $G$ is a connected graph and $T$ is a spanning tree of $G$ . If $(G, T)$ is a ct-pair, then by...

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...

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...

A partition of the Catalan numbers and enumeration of genealogical trees

Rainer Schimming (1996)

Discussiones Mathematicae Graph Theory

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A special relational structure, called genealogical tree, is introduced; its social interpretation and geometrical realizations are discussed. The numbers $C_{n, k}$ of all abstract genealogical trees with exactly n+1 nodes and k leaves is found by means of enumeration of code words. For each n, the $C_{n, k}$ form a partition of the n-th Catalan numer Cₙ, that means $C_{n, 1} + C_{n, 2} + . . . + C_{n, n} = C ₙ$ .

On operators from separable reflexive spaces with asymptotic structure

Bentuo Zheng (2008)

Studia Mathematica

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Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower- $ℓ_{q}$ -tree estimate and let T be a bounded linear operator from X which satisfies an upper- $ℓ_{p}$ -tree estimate. Then T factors through a subspace of ${(\sum F ₙ)}_{ℓ_{r}}$ , where (Fₙ) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an $(ℓ_{p}, ℓ_{q})$ FDD. Similarly, let 1 < q < r < p < ∞ and let X be a separable reflexive Banach space satisfying an asymptotic...

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.

Note about sequences of extrema $(A, 2 B)$ -edge coloured trees

Urszula Bednarz, Iwona Włoch (2017)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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In this paper we determine successive extremal trees with respect to the number of all $(A, 2 B)$ -edge colourings.

Compactness properties of weighted summation operators on trees

Mikhail Lifshits, Werner Linde (2011)

Studia Mathematica

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We investigate compactness properties of weighted summation operators $V_{α, σ}$ as mappings from ℓ₁(T) into $ℓ_{q} (T)$ for some q ∈ (1,∞). Those operators are defined by $(V_{α, σ} x) (t) : = α (t) \sum_{s ⪰ t} σ (s) x (s)$ , t ∈ T, where T is a tree with partial order ⪯. Here α and σ are given weights on T. We introduce a metric d on T such that compactness properties of (T,d) imply two-sided estimates for $e ₙ (V_{α, σ})$ , the (dyadic) entropy numbers of $V_{α, σ}$ . The results are applied to concrete trees, e.g. moderately increasing, biased or binary trees and to weights...

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...

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.