Properties of Subtree-Prune-and-Regraft Operations on Totally-Ordered Phylogenetic Trees

We study some properties of subtree-prune-and-regraft (SPR) operations on leaflabelled rooted binary trees in which internal vertices are totally ordered. Since biological events occur with certain time ordering, sometimes such totally-ordered trees must be used to avoid possible contradictions in representing evolutionary histories of biological sequences. Compared to the case of plain leaf-labelled rooted binary trees where internal vertices are only partially ordered, SPR operations on totally-ordered trees are more constrained and therefore more difficult to study. In this paper, we investigate the unit-neighbourhood U(T), defined as the set of totally-ordered trees one SPR operation away from a given totally-ordered tree T. We construct a recursion relation for | U(T) | and thereby arrive at an efficient method of determining | U(T) |. In contrast to the case of plain rooted trees, where the unit-neighbourhood size grows quadratically with respect to the number n of leaves, for totally-ordered trees | U(T) | grows like O(n³). For some special topology types, we are able to obtain simple closed-form formulae for | U(T) |. Using these results, we find a sharp upper bound on | U(T) | and conjecture a formula for a sharp lower bound. Lastly, we study the diameter of the space of totally-ordered trees measured using the induced SPR-metric. Received May 18, 2004     

Some results on spanning trees

Some structures of spanning trees with many or less leaves in a connected graph are determined.We show（1） a connected graph G has a spanning tree T with minimum leaves such that T contains a longest path,and（2） a connected graph G on n vertices contains a spanning tree T with the maximum leaves such that Δ（G） =Δ（T） and the number of leaves of T is not greater than n D（G）＋1,where D（G） is the diameter of G.     

Smith Normal Form and acyclic matrices

An approach, based on the Smith Normal Form, is introduced to study the spectra of symmetric matrices with a given graph. The approach serves well to explain how the path cover number (resp. diameter of a tree T) is related to the maximal multiplicity MaxMult(T) occurring for an eigenvalue of a symmetric matrix whose graph is T (resp. the minimal number q(T) of distinct eigenvalues over the symmetric matrices whose graphs are T). The approach is also applied to a more general class of connected graphs G, not necessarily trees, in order to establish a lower bound on q(G).     

Low-Light Trees, and Tight Lower Bounds for Euclidean Spanners

We show that for every n-point metric space M and positive integer k, there exists a spanning tree T with unweighted diameter O(k) and weight w(T)=O(k⋅n ^1/k )⋅w(MST(M)), and a spanning tree T′ with weight w(T′)=O(k)⋅w(MST(M)) and unweighted diameter O(k⋅n ^1/k ). These trees also achieve an optimal maximum degree. Furthermore, we demonstrate that these trees can be constructed efficiently.     

Bipartite labelings of trees and the gracesize

Let T = (V, E) be a tree whose vertices are properly 2-colored. A bipartite labeling of T is a bijection f: V ← {0, 1, ?, | E |} for which there is a k such that whenever f(u) ≤ k < f(v), then u and v have different colors. The α-size of the tree T is the maximum number of distinct values of the induced edge labels |f(u) - f(v)|, uv ? E, taken over all bipartite labelings f of T. We investigate the asymptotic behavior of the α-size of trees. Let α(n) be the smallest α-size among all the trees with n edges. As our main result we prove that 5(n + 1)/7 ≤ α(n) ≤ (5n + 9)/6. A connection with the graceful tree conjecture is established, in that every tree with n edges is shown to have “gracesize” at least 5n/7. © 1995 John Wiley & Sons, Inc.     

Bipartite labeling of trees with maximum degree three

Let T = (V, E) be a tree with a properly 2‐colored vertex set. A bipartite labeling of T is a bijection φ: V → {1, …, |V|} for which there exists a k such that whenever φ(u) ≤ k < φ(v), then u and v have different colors. The α‐size α(T) of the tree T is the maximum number of elements in the sets {|φ(u) − φ(v)|; uv ∈ E}, taken over all bipartite labelings φ of T. The quantity α(n) is defined as the minimum of α(T) over all trees with n vertices. In an earlier article (J Graph Theory 19 (1995), 201–215), A. Rosa and the second author proved that 5n/7 ≤ α(n) ≤ (5n + 4)/6 for all n ≥ 4; the upper bound is believed to be the asymptotically correct value of (n). In this article, we investigate the α‐size of trees with maximum degree three. Let α₃(n) be the smallest α‐size among all trees with n vertices, each of degree at most three. We prove that α₃(n) ≥ 5n/6 for all n ≥ 12, thus supporting the belief above. This result can be seen as an approximation toward the graceful tree conjecture—it shows that every tree on n ≥ 12 vertices and with maximum degree three has “gracesize” at least 5n/6. Using a computer search, we also establish that α₃(n) ≥ n − 2 for all n ≤ 17. © 1999 John Wiley & Sons, Inc. J Graph Theory 31:7–15, 1999     

Null decomposition of trees

Let $T$ be a tree. We show that the null space of the adjacency matrix of $T$ has relevant information about the structure of $T$. We introduce the Null Decomposition of trees, which is a decomposition into two different types of trees: N-trees and S-trees. N-trees are the trees that have a unique maximum (perfect) matching. S-trees are the trees with a unique maximum independent set. We obtain formulas for the independence number and the matching number of a tree using this decomposition. We also show how the number of maximum matchings and the number of maximum independent sets in a tree are related to its null decomposition.     

Embeddings and Other Mappings of Rooted Trees Into Complete Trees

Let T ⁿ be the complete binary tree of height n, with root 1 _n as the maximum element. For T a tree, define and . We disprove a conjecture of Kubicki, Lehel and Morayne, which claims that for any fixed n and arbitrary rooted trees T ₁ T ₂. We show that A(n; T) is of the form where l is the number of leaves of T, and each q _j is a polynomial. We provide an algorithm for calculating the two leading terms of q _l for any tree T. We investigate the asymptotic behaviour of the ratio A(n; T)/C(n; T) and give examples of classes of pairs of trees T ₁, T ₂ where it is possible to compare A(n; T ₁)/C(n; T ₁) and A(n; T ₂)/C(n; T ₂). By calculating these ratios for a particular class of pairs of trees, we show that the conjecture fails for these trees, for all sufficiently large n. Kubicki, Lehel and Morayne have proved the conjecture when T ₁, T ₂ are restricted to being binary trees. We also look at embeddings into other complete trees, and we show how the result can be viewed as one of many possible correlation inequalities for embeddings of binary trees. We also show that if we consider strict order-preserving maps of T ₁, T ₂ into T ⁿ (rather than embeddings) then the corresponding correlation inequalities for these maps also generalise to arbitrary trees.     

Convex labelings of trees

A convex labeling of a tree T of order n is a one-to-one function f from the vertex set of T into the nonnegative integers, so that f(y) ? (f(x) + f(z))/2 for every path x, y, z of length 2 in T. If, in addition, f(v) ? n ? 1 for every vertex v of T, then f is a perfect convex labeling and T is called a perfectly convex tree. Jamison introduced this concept and conjectured that every tree is perfectly convex. We show that there exists an infinite class of trees, none of which is perfectly convex, and in fact prove that for every n there exists a tree of order n which requires a convex labeling with maximum value at least 6n/5 – 22. We also prove that every tree of order n admits a convex labeling with maximum label no more than n²/8 + 2. In addition, we present some constructive methods for obtaining perfect convex labelings of large classes of trees.     

The 2-intersection number of paths and bounded-degree trees

We represent a graph by assigning each vertex a finite set such that vertices are adjacent if and only if the corresponding sets have at least two common elements. The 2-intersection number θ₂(G) of a graph G is the minimum size of the union of sets in such a representation. We prove that the maximum order of a path that can be represented in this way using t elements is between (t² - 19t + 4)/4 and (t² - t + 6)/4, making θ₂(P_n) asymptotic to 2√n. We also show the existence of a constant c depending on ? such that, for any tree T with maximum degree at most d, θ₂(T) ≤ c(√n)^1+?. When the maximum degree is not bounded, there is an n-vertex tree T with θ₂(T) > .945n^2/3. © 1995 John Wiley & Sons, Inc.     

Perfect Matchings of Cellular Graphs

We introduce a family of graphs, called cellular, and consider the problem of enumerating their perfect matchings. We prove that the number of perfect matchings of a cellular graph equals a power of 2 times the number of perfect matchings of a certain subgraph, called the core of the graph. This yields, as a special case, a new proof of the fact that the Aztec diamond graph of order n introduced by Elkies, Kuperberg, Larsen and Propp has exactly 2 ^n(n+1)/2 perfect matchings. As further applications, we prove a recurrence for the number of perfect matchings of certain cellular graphs indexed by partitions, and we enumerate the perfect matchings of two other families of graphs called Aztec rectangles and Aztec triangles.     

UPPER BOUNDS ON THE LOWER OPEN PACKING NUMBER OF A TREE

Abstract

Let G be a graph and let v be a vertex of G. The open neigbourhood N(v) of v is the set of all vertices adjacent with v in G. An open packing of G is a set of vertices whose open neighbourhoods are pairwise disjoint. The lower open packing number of G, denoted ρ^° _L(G), is the minimum cardinality of a maximal open packing of G while the (upper) open packing number of G, denoted ρ^°(G), is the maximum cardinality among all open packings of G. It is known (see [7]) that if G is a connected graph of order n ≥3, then ρ^°(G) ≤ 2n/3 and this bound is sharp (even for trees). As a consequence of this result, we know that ρ^° _L(G) ≤ 2n/3. In this paper, we improve this bound when G is a tree. We show that if G is a tree of order n with radius 3, then ρ^° _L(G) ≤ n/2 + 2 √n-1, and this bound is sharp, while if G is a tree of order n with radius at least 4, then ρ^° _L(G) is bounded above by 2n/3—O√n).     

The spanning trees forced by the path and the star

A set S of trees of order n forces a tree T if every graph having each tree in S as a spanning tree must also have T as a spanning tree. A spanning tree forcing set for order n that forces every tree of order n. A spanning-tree forcing set S is a test set for panarboreal graphs, since a graph of order n is panarboreal if and only if it has all of the trees in S as spanning trees. For each positive integer n ≠ 1, the star belongs to every spanning tree forcing set for order n. The main results of this paper are a proof that the path belongs to every spanning-tree forcing set for each order n ∉ {1, 6, 7, 8} and a computationally tractable characterization of the trees of order n ≥ 15 forced by the path and the star. Corollaries of those results include a construction of many trees that do not belong to any minimal spanning tree forcing set for orders n ≥ 15 and a proof that the following related decision problem is NP-complete: an instance is a pair (G, T) consisting of a graph G of order n and maximum degree n - 1 with a hamiltonian path, and a tree T of order n; the problem is to determine whether T is a spanning tree of G. © 1996 John Wiley & Sons, Inc.     

Ordering Trees with Nearly Perfect Matchings by Algebraic Connectivity

Let T2k＋1 be the set of trees on 2k＋1 vertices with nearly perfect matchings and α（T） be the algebraic connectivity of a tree T. The authors determine the largest twelve values of the algebraic connectivity of the trees in T2k＋1. Specifically, 10 trees T2,T3,... ,T11 and two classes of trees T（1） and T（12） in T2k＋1 are introduced. It is shown in this paper that for each tree T^′1,T^″1∈T（1）and T^′12,T^″12∈T（12） and each i,j with 2≤i〈j≤11,α（T^′1）=α（T^″1）〉α（Tj）〉α（T^′12）=α（T^″12）.It is also shown that for each tree T with T∈T2k＋1／（T（1）∪{T2,T3,…,T11}∪T（12））,α（T^′12）〉α（T）.     

A parallel algorithm for generating multiple ordering spanning trees in undirected weighted graphs

1.IntroductionLetG=(V,E,W)beaconnected,weightedandundirectedgraph,VeEE,w(e)(相似文献   

Embedding Tree Metrics into Low-Dimensional Euclidean Spaces

We consider embedding metrics induced by trees into Euclidean spaces with a restricted number of dimensions. We show that any weighted tree T with n vertices and L leaves can be embedded into d -dimensional Euclidean space with ? (L ^1/(d-1) ) distortion. Furthermore, we exhibit an embedding with almost the same distortion which can be computed efficiently. This distortion substantially improves the previous best upper bound of \tilde O (n ^2/d ) and almost matches the best known lower bound of Ω(L ^1/d ) . Received August 17, 1999, and in revised form January 25, 2000.     

Extremal energies of trees with a given domination number

The energy of a graph is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. Let T(n,γ) be the set of trees of order n and with domination number γ. In this paper, we characterize the tree from T(n,γ) with the minimal energy, and determine the tree from T(n,γ) where n=kγ with maximal energy for .     

Bounds on the number of maximal sum-free sets

We show that the number of maximal sum-free subsets of {1,2,…,n} is at most 2^3n/8+o(n). We also show that 2^0.406n+o(n) is an upper bound on the number of maximal product-free subsets of any group of order n.     

Bounds on the convex label number of trees

A convex labelling of a tree is an assignment of distinct non-negative integer labels to vertices such that wheneverx, y andz are the labels of vertices on a path of length 2 theny≦(x+z)/2. In addition if the tree is rooted, a convex labelling must assign 0 to the root. The convex label number of a treeT is the smallest integerm such thatT has a convex labelling with no label greater thanm. We prove that every rooted tree (and hence every tree) withn vertices has convex label number less than 4n. We also exhibitn-vertex trees with convex label number 4n/3+o(n), andn-vertex rooted trees with convex label number 2n +o(n). The research by M. B. and A. W. was partly supported by NSF grant MCS—8311422.     

Polynomial area bounds for MST embeddings of trees

In their seminal paper on geometric minimum spanning trees, Monma and Suri (1992) [31] showed how to embed any tree of maximum degree 5 as a minimum spanning tree in the Euclidean plane. The embeddings provided by their algorithm require area O(n²²)×O(n²²) and the authors conjectured that an improvement below cn×cn is not possible, for some constant c>0. In this paper, we show how to construct MST embeddings of arbitrary trees of maximum degree 3 and 4 within polynomial area.