Graphs of Triangulations and Perfect Matchings

Houle, Michael E.; Hurtado, Ferrán; Noy, Marc; Rivera-Campo, Eduardo

doi:10.1007/s00373-005-0615-2

Cited by 26 publications

(16 citation statements)

References 7 publications

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“…There it is shown that a sequence of length O (log n) suffices to transform any perfect matching into another perfect matching on a given set of n = 2m points. This improves the previously best-known linear upper bound of n −2 by Houle et al [9], who also showed that the corresponding transformation graph is connected. Recently, Razen [10] proved a corresponding lower bound of Ω(log n/ log log n) with methods similar to the ones used in this paper.…”

supporting

confidence: 63%

Transforming spanning trees: A lower bound

Buchin

Razen

Uno

et al. 2009

Computational Geometry

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For a planar point set we consider the graph whose vertices are the crossing-free straightline spanning trees of the point set, and two such spanning trees are adjacent if their union is crossing-free. An upper bound on the diameter of this graph implies an upper bound on the diameter of the flip graph of pseudo-triangulations of the underlying point set.We prove a lower bound of Ω(log n/ log log n) for the diameter of the transformation graph of spanning trees on a set of n points in the plane. This nearly matches the known upper bound of O (log n). If we measure the diameter in terms of the number of convex layers k of the point set, our lower bound construction is tight, i.e., the diameter is in Ω(log k) which matches the known upper bound of O (log k). So far only constant lower bounds were known.

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supporting

confidence: 63%

Transforming spanning trees: A lower bound

Buchin

Razen

Uno

et al. 2009

Computational Geometry

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“…, T k−1 , T k = T b , with k ∈ O(log n), in which every two consecutive trees are compatible [2]. Similar results have been obtained for other configurations, such as matchings [3,15], and (in the negative) for pointed pseudotriangulations [4]. A major open problem in the area is whether this kind of sequences exist or not for polygonizations, or for spanning paths, of any given point set.…”

Section: Introductionsupporting

confidence: 51%

Compatible spanning trees

García

Huemer

Hurtado

et al. 2014

Computational Geometry

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Two plane geometric graphs are said to be compatible when their union is a plane geometric graph. Let S be a set of n points in the Euclidean plane in general position and let T be any given plane geometric tree whose vertex set is S. The main problem we consider in this work consists of finding a second plane geometric tree T on S, such that T is compatible with T and shares with T a minimum number of edges. We prove, up to additive constants, that there is always a compatible plane geometric tree T having in common with T at most n/4 edges; while for some plane geometric trees T , any other tree T spanning S, compatible with T , has at least n/5 edges in common with T .

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“…Moreover, from the viewpoint of the applications to enumeration problems, the transformation should be defined locally, i.e., the symmetric difference between two connected objects should be as small as possible. The design of the transformation for a set of noncrossing graphs might be interesting in its own right, and there are many known results not only for the local transformations [22,20,19,2,5,6,18] but also for the large transformations [1,3,21]. In fact, almost all previous works for the enumeration discussed above are based on the local transformations.…”

Section: General Ideamentioning

confidence: 96%

Fast enumeration algorithms for non-crossing geometric graphs

Katoh

Tanigawa

2008

Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry

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A non-crossing geometric graph is a graph embedded on a given set of points in the plane with non-crossing straight line segments. In this paper we present a new general framework for enumerating non-crossing geometric graphs for a given point set. By applying our idea to specific enumeration problems, we obtain faster algorithms for enumerating plane straight-line graphs, non-crossing spanning connected graphs, non-crossing spanning trees and non-crossing minimally rigid frameworks. Furthermore, we also obtain efficient enumeration algorithms for non-crossing geometric graph classes, for which no enumeration algorithm has been reported so far, such as non-crossing matchings, noncrossing blue-and-red matchings, non-crossing k-vertex or k-edge connected graphs or non-crossing directed spanning trees. The proposed idea is relatively simple, and can be potentially applied to various other enumeration problems of non-crossing geometric graphs.

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Graphs of Triangulations and Perfect Matchings

Cited by 26 publications

References 7 publications

Transforming spanning trees: A lower bound

Transforming spanning trees: A lower bound

Compatible spanning trees

Fast enumeration algorithms for non-crossing geometric graphs

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