Preface

The concept of tree (in a graph) has proven to be fundamental in several areas of applied mathematics such as graph theoryand operations research. This issue confirms this fact and shows new areas of application where trees play a pivotal role.Among others, we mention the interdisciplinary application (involving the areas of optimization and biology) that focuseson the determination of so-called phylogenetic trees. Other applications involve emergency evacuation networks and moreconventional telecommunications settings. From a theoretical point of view, this special issue contains contributions that (i)study a graph theoretical problem posed on a tree network or (ii) efficiently find a tree with a special cost structure.

Below, we provide a brief description of the papers constituting this issue.

Andreas and Smith [1] describe the design of an evacuation tree subject to capacity constraints on the links. The authorspropose a solution method based on Benders decomposition and test the method on randomly generated instances with up to10 nodes and 10 different disaster scenarios.

Brazil, et al. [2] study the determination of phylogenetic trees and propose a new distance-based heuristic for their deter-mination. The principal and novel idea of the proposed heuristic is to view the problem of locating species in d-dimensionalspace and then exploit concepts of Euclidean geometry to build a low-cost Steiner tree in d-space.

Catanzaro [3] proposes an overview of the minimum evolution problem that characterizes one of the established criteriafor determining phylogenetic trees. This overview focus on variations and methods to solve this important problem.

Catanzaro, et al. [4] propose, examine, and test several mathematical models to construct phylogenetic trees under theminimum evolution criterion. This study involves describing models in several different variable spaces as well as proposingvalid inequalities to enhance the linear programming relaxation of the proposed models. Computational results are presentedfor instances with up to 10 species for which the optimal solutions are obtained.

The paper by Costa, et al. [5] describes a variation of the hop-constrained Steiner tree problem with two additional features:(i) revenues are associated with the nodes and (ii) a budget constraint is imposed specifying an upper limit on the cost of thenetwork. The authors describe and compare variants of the so-called hop-indexed models and develop a branch-and-cut modelthat solves to optimality instances with up to 500 nodes and 625 edges.

Dabney, et al. [6] discuss a variant of the classic minimum dominating set problem on a tree network. They propose a newalgorithm for the problem when defined on a tree and which improves upon the complexity of the best previously knownalgorithms.

Drummond, et al. [7] propose a distributed version of Wong’s dual ascent algorithm for a distributed version of the minimumSteiner tree problem that arises in applications such as multicasting. Computational results show the effectiveness of the newapproach.

Goddard and Hedetniemi [8] investigate the properties of and operations on so-called tables that represent the state space ofmany algorithms that use recursively decomposable data structures. The paper focuses on various graph algorithms for trees.

Köhler, et al. [9] study the problem of computing a spanning tree that minimizes the sum of the lengths of its inducedcycles. The authors propose new (and effective) dual bounds for the problem defined on planar grid graphs.

Received April 2008; accepted June 2008Correspondence to: L. Gouveia; University of Lisbon, Faculty of Sciences, Department of Statistics and Operations Research, Cidade Universitaria, BlocoC6, Campo Grande, Lisbon, 1749-016 Portugal; e-mail: legouveia@fc.ul.ptDOI 10.1002/net.20277Published online 22 October 2008 in Wiley InterScience (www.interscience.wiley.com).© 2008 Wiley Periodicals, Inc.

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Laskar, et al. [10] study a variant of a coloring problem on trees that arises in channel assignment applications. The authorsshow that if the given graph is a tree other than a star, then it can be L(2,1)-colored in a particular way.

Prendergast, et al. [11] study the behavior of the Steiner ratios for all configurations connecting 3 points in space.

Puerto, et al. [12] consider a location problem with path-shaped facilities on a tree network. The authors consider theobjective of minimizing the variance and discuss the continuous as well as the discrete cases.

Luis GouveiaGuest Editor

REFERENCES

[1] A. Andreas and J.C. Smith, Decomposition algorithms for the design of a non-simultaneous capacitated evacuation tree network,Networks 53 (2009), 91–103.

[2] M. Brazil, B. Nielsen, D. Thomas, P. Winter, C. Wulff-Nielsen, and M. Zachariasen, A novel approach to phylogenetic trees:d-dimensional geometric Steiner trees, Networks 53 (2009), 104–111.

[3] D. Catanzaro, The minimum evolution problem: Overview and classification, Networks 53 (2009), 112–125.

[4] D. Catanzaro, M. Labbé, R. Pesenti, and J.-J. Salazar-González, Mathematical models to reconstruct phylogenetic trees under theminimum evolution criterion, Networks 53 (2009), 126–140.

[5] A. Costa, J.-F. Cordeau, and G. Laporte, Models and branch-and-cut algorithms for the Steiner tree problem with revenues, budgetand hop constraints, Networks 53 (2009), 141–159.

[6] J. Dabney, B. Dean, and S. Hedetniemi, A linear-time algorithm for broadcast domination in a tree, Networks 53 (2009), 160–169.

[7] L. Drummond, M. Santos, and E. Uchoa, A distributed dual ascent algorithm for Steiner problems in multicast routing, Networks 53(2009), 170–183.

[8] W. Goddard and S. Hedetniemi, A note on trees, tables and algorithms, Networks 53 (2009), 184–190.

[9] E. Köhler, C. Liebchen, R. Rizzi, and G. Wunsch, Lower bounds for strictly fundamental cycle bases in grid graphs, Networks 53(2009), 191–205.

[10] R. Laskar, G. Matthews, B. Novick, and J. Villalpando, On irreducible no-hole L(2,1)-coloring of trees, Networks 53 (2009), 206–211.

[11] K. Prendergast, D. Thomas, and J. Weng, Optimum Steiner ratio for gradient-constrained networks connecting three points in 3-space,Networks 53 (2009), 212–220.

[12] J. Puerto, F. Ricca, and A. Scozzari, The continuous and discrete path-variance problems on trees, Networks 53 (2009), 221–228.

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