A substitution theorem for graceful trees and its applications

A graceful labeling of a graph G=(V,E) assigns |V| distinct integers from the set {0,…,|E|} to the vertices of G so that the absolute values of their differences on the |E| edges of G constitute the set {1,…,|E|}. A graph is graceful if it admits a graceful labeling. The forty-year old Graceful Tree Conjecture, due to Ringel and Kotzig, states that every tree is graceful.We prove a Substitution Theorem for graceful trees, which enables the construction of a larger graceful tree through combining smaller and not necessarily identical graceful trees. We present applications of the Substitution Theorem, which generalize earlier constructions combining smaller trees.     

Enumerating spanning trees of graphs with an involution

As the extension of the previous work by Ciucu and the present authors [M. Ciucu, W.G. Yan, F.J. Zhang, The number of spanning trees of plane graphs with reflective symmetry, J. Combin. Theory Ser. A 112 (2005) 105-116], this paper considers the problem of enumeration of spanning trees of weighted graphs with an involution which allows fixed points. We show that if G is a weighted graph with an involution, then the sum of weights of spanning trees of G can be expressed in terms of the product of the sums of weights of spanning trees of two weighted graphs with a smaller size determined by the involution of G. As applications, we enumerate spanning trees of the almost-complete bipartite graph, the almost-complete graph, the Möbius ladder, and the almost-join of two copies of a graph.     

Maximizing the minimum voter satisfaction on spanning trees

This paper analyzes the computational complexity involved in solving fairness issues on graphs, e.g., in the installation of networks such as water networks or oil pipelines. Based on individual rankings of the edges of a graph, we will show under which conditions solutions, i.e., spanning trees, can be determined efficiently given the goal of maximin voter satisfaction. In particular, we show that computing spanning trees for maximin voter satisfaction under voting rules such as approval voting or the Borda count is -complete for a variable number of voters whereas it remains polynomially solvable for a constant number of voters.     

Inductive item tree analysis: Corrections, improvements, and comparisons

There are various methods in knowledge space theory for building knowledge structures or surmise relations from data. Few of them have been thoroughly analyzed, making it difficult to decide which of these methods provides good results and when to apply each of the methods.In this paper, we investigate the method known as inductive item tree analysis and discuss the advantages and disadvantages of this algorithm. In particular, we introduce some corrections and improvements to it, resulting in two newly proposed algorithms. These algorithms and the original inductive item tree analysis procedure are compared in a simulation study and with empirical data.     

On the approximability of the Maximum Agreement SubTree and Maximum Compatible Tree problems

The aim of this paper is to give a complete picture of approximability for two tree consensus problems which are of particular interest in computational biology: Maximum Agreement SubTree (MAST) and Maximum Compatible Tree (MCT). Both problems take as input a label set and a collection of trees whose leaf sets are each bijectively labeled with the label set. Define the size of a tree as the number of its leaves. The well-known MAST problem consists of finding a maximum-sized tree that is topologically embedded in each input tree, under label-preserving embeddings. Its variant MCT is less stringent, as it allows the input trees to be arbitrarily refined. Our results are as follows. We show that MCT is -hard to approximate within bound n^1−? on rooted trees, where n denotes the size of each input tree; the same approximation lower bound was already known for MAST [J. Jansson, Consensus algorithms for trees and strings, Ph. D. Thesis, Lund University, 2003]. Furthermore, we prove that MCT on two rooted trees is not approximable within bound 2^log1−?n, unless all problems in are solvable in quasi-polynomial time; the same result was previously established for MAST on three rooted trees [J. Hein, T. Jiang, L. Wang, K. Zhang, On the complexity of comparing evolutionary trees, Discrete Applied Mathematics 71 (1-3) (1996) 153-169] (note that MAST on two trees is solvable in polynomial time [M.A. Steel, T.J. Warnow, Kaikoura tree theorems: Computing the maximum agreement subtree, Information Processing Letters 48 (2) (1993) 77-82]). Let CMAST, resp. CMCT, denote the complement version of MAST, resp. MCT: CMAST, resp. CMCT, consists of finding a tree that is a feasible solution of MAST, resp. MCT, and whose leaf label set excludes a smallest subset of the input labels. The approximation threshold for CMAST, resp. CMCT, on rooted trees is shown to be the same as the approximation threshold for CMAST, resp. CMCT, on unrooted trees; it was already known that both CMAST and CMCT are approximable within ratio three on rooted and unrooted trees [V. Berry, F. Nicolas, Maximum agreement and compatible supertrees, in: S.C. Sahinalp, S. Muthukrishnan, U. Dogrusoz (Eds.), Proceedings of the 15th Annual Symposium on Combinatorial Pattern Matching, CPM’04, in: LNCS, vol. 3109, Springer-Verlag, 2004, pp. 205-219; G. Ganapathy, T.J. Warnow, Approximating the complement of the maximum compatible subset of leaves of k trees, in: K. Jansen, S. Leonardi, V. V. Vazirani (Eds.), Proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization, APPROX’02, in: LNCS, vol. 2462, Springer-Verlag, 2002, pp. 122-134]. The latter results are completed by showing that CMAST is -hard on three rooted trees and that CMCT is -hard on two rooted trees.     

Shortest paths in linear time on minor-closed graph classes, with an application to Steiner tree approximation

We generalize the linear-time shortest-paths algorithm for planar graphs with nonnegative edge-weights of Henzinger et al. (1994) to work for any proper minor-closed class of graphs. We argue that their algorithm can not be adapted by standard methods to all proper minor-closed classes. By using recent deep results in graph minor theory, we show how to construct an appropriate recursive division in linear time for any graph excluding a fixed minor and how to transform the graph and its division afterwards, so that it has maximum degree three. Based on such a division, the original framework of Henzinger et al. can be applied. Afterwards, we show that using this algorithm, one can implement Mehlhorn’s (1988) 2-approximation algorithm for the Steiner tree problem in linear time on these graph classes.     

四维余代数的分类

利用余代数的树结构基,得到了余代数同构的等价条件,从而完成了四维余代数的分类,并针对更高维的余代数分类给出了一般方法.     

Layers and matroids for the traveling salesman’s paths

Gottschalk and Vygen proved that every solution of the subtour elimination linear program for traveling salesman paths is a convex combination of more and more restrictive “generalized Gao-trees”. We give a short proof of this fact, as a layered convex combination of bases of a sequence of increasingly restrictive matroids. A strongly polynomial, combinatorial algorithm follows for finding this convex combination, which is a new tool offering polyhedral insight, already instrumental in recent results for the $s?t$ path TSP.     

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Take any subcritical offspring distribution with bounded support and consider the corresponding Galton--Watson tree. In this short note we condition this Galton--Watson tree on large width and show that the conditioned tree does not converge locally to any random tree with at most one infinite spine.