Signpost systems and spanning trees of graphs

Ladislav Nebeský

Displaying similar documents to “Signpost systems and spanning trees of graphs”

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...

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.

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...

The induced paths in a connected graph and a ternary relation determined by them

Ladislav Nebeský (2002)

Mathematica Bohemica

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By a ternary structure we mean an ordered pair $(X_{0}, T_{0})$ , where $X_{0}$ is a finite nonempty set and $T_{0}$ is a ternary relation on $X_{0}$ . By the underlying graph of a ternary structure $(X_{0}, T_{0})$ we mean the (undirected) graph $G$ with the properties that $X_{0}$ is its vertex set and distinct vertices $u$ and $v$ of $G$ are adjacent if and only if ${x \in X_{0} T_{0} (u, x, v)} \cup {x \in X_{0} T_{0} (v, x, u)} = {u, v} .$ A ternary structure $(X_{0}, T_{0})$ is said to be the B-structure of a connected graph $G$ if $X_{0}$ is the vertex set of $G$ and the following statement holds for all $u, x, y \in X_{0}$ : $T_{0} (x, u, y)$ if and only if $u$ belongs to an...

The Wiener number of powers of the Mycielskian

Rangaswami Balakrishnan, S. Francis Raj (2010)

Discussiones Mathematicae Graph Theory

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The Wiener number of a graph G is defined as $1 / 2 \sum_{u, v \in V (G)} d (u, v)$ , d the distance function on G. The Wiener number has important applications in chemistry. We determine a formula for the Wiener number of an important graph family, namely, the Mycielskians μ(G) of graphs G. Using this, we show that for k ≥ 1, $W (μ (S ₙ^{k})) \leq W (μ (T ₙ^{k})) \leq W (μ (P ₙ^{k}))$ , where Sₙ, Tₙ and Pₙ denote a star, a general tree and a path on n vertices respectively. We also obtain Nordhaus-Gaddum type inequality for the Wiener number of $μ (G^{k})$ .

On the spectral radius of $‡$ -shape trees

Xiaoling Ma, Fei Wen (2013)

Czechoslovak Mathematical Journal

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Let $A (G)$ be the adjacency matrix of $G$ . The characteristic polynomial of the adjacency matrix $A$ is called the characteristic polynomial of the graph $G$ and is denoted by $φ (G, λ)$ or simply $φ (G)$ . The spectrum of $G$ consists of the roots (together with their multiplicities) $λ_{1} (G) \geq λ_{2} (G) \geq ... \geq λ_{n} (G)$ of the equation $φ (G, λ) = 0$ . The largest root $λ_{1} (G)$ is referred to as the spectral radius of $G$ . A $‡$ -shape is a tree with exactly two of its vertices having maximal degree 4. We will denote by $G (l_{1}, l_{2}, ..., l_{7})$ $(l_{1} \geq 0$ , $l_{i} \geq 1$ , $i = 2, 3, ..., 7)$ a $‡$ -shape tree such that...

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.

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...

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.

Spanning caterpillars with bounded diameter

Ralph Faudree, Ronald Gould, Michael Jacobson, Linda Lesniak (1995)

Discussiones Mathematicae Graph Theory

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A caterpillar is a tree with the property that the vertices of degree at least 2 induce a path. We show that for every graph G of order n, either G or G̅ has a spanning caterpillar of diameter at most 2 log n. Furthermore, we show that if G is a graph of diameter 2 (diameter 3), then G contains a spanning caterpillar of diameter at most $c n^{3 / 4}$ (at most n).