Partitions of complete geometric graphs into plane trees

Bose, Prosenjit; Hurtado, Ferrán; Rivera-Campo, Eduardo; Wood, David R.

doi:10.1016/j.comgeo.2005.08.006

Cited by 33 publications

(38 citation statements)

References 12 publications

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“…It is often useful to restrict the subgraphs of G to a certain class or property. Among all subgraphs of K n , plane spanning trees, plane Hamiltonian cycles or paths, and plane perfect matchings, are of interest [1][2][3]8] i.e., one may look for the maximum number of these subgraphs that can be packed into K n .…”

Section: Introductionmentioning

confidence: 99%

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Packing 1-plane Hamiltonian cycles in complete geometric graphs

Michman

Ali

Chia

et al. 2019

Filomat

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Counting the number of Hamiltonian cycles that are contained in a geometric graph is #Pcomplete even if the graph is known to be planar. A relaxation for problems in plane geometric graphs is to allow the geometric graphs to be 1-plane, that is, each of its edges is crossed at most once. We consider the following question: For any set P of n points in the plane, how many 1-plane Hamiltonian cycles can be packed into a complete geometric graph K n ? We investigate the problem by taking three different situations of P, namely, when P is in convex position and when P is in wheel configurations position. Finally, for points in general position we prove the lower bound of k − 1 where n = 2 k + h and 0 ≤ h < 2 k. In all of the situations, we investigate the constructions of the graphs obtained.

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Section: Introductionmentioning

confidence: 99%

“…Bernhart and Kanien [5] give an affirmative answer for the problem when the points are in convex position. Bose et al [8] proved that every complete geometric graph K n can be partitioned into at most n − n 12 plane trees. Aichholzer et al [3] showed that Ω( √ n) plane spanning trees can be packed into K n .…”

Section: Introductionmentioning

confidence: 99%

Packing 1-plane Hamiltonian cycles in complete geometric graphs

Michman

Ali

Chia

et al. 2019

Filomat

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“…Let us start right away with an illustrating example, which did indeed motivate our study. We came across the following intriguing open question, which was considered in [11] and investigated further in [6] and [24]: Given a complete geometric graph (edges as straight segments) on 2m points in general position in the plane, is it always possible to partition the edges into m crossing-free spanning trees? For addressing such problems, the concept of order types 1 is ubiquitously used in discrete and computational geometry, as it allows for classifying the infinite number of point sets of a given size into a finite number of equivalence classes, capturing combinatorial properties such as which pairs of spanned line segments cross and which points define the set's convex hull.…”

Section: Introductionmentioning

confidence: 99%

Order on Order Types

Pilz

Welzl

2017

Discrete Comput Geom

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Given P and P , equally sized planar point sets in general position, we call a bijection from P to P crossing-preserving if crossings of connecting segments in P are preserved in P (extra crossings may occur in P ). If such a mapping exists, we say that P crossing-dominates P, and if such a mapping exists in both directions, P and P are called crossing-equivalent. The relation is transitive, and we have a partial order on the obtained equivalence classes (called crossing types or x-types). Point sets of equal order type are clearly crossing-equivalent, but not vice versa. Thus, x-types are a coarser classification than order types. (We will see, though, that a collapse of different order types to one x-type occurs for sets with triangular convex hull only.) We argue that either the maximal or the minimal x-types are sufficient for answering many combinatorial (existential or extremal) questions on planar point sets. Motivated by this we consider basic properties of the relation. We characterize order types crossingdominated by points in convex position. Further, we give a full characterization of minimal and maximal abstract order types. Based on that, we provide a polynomialtime algorithm to check whether a point set crossing-dominates another. Moreover, we generate all maximal and minimal x-types for small numbers of points.Dedicated to Jacob E. Goodman and Richard Pollack on the occasion of their eightieth birthdays.

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“…It is unknown whether for some constant ε > 0, every geometric drawing of K n has thickness at most (1 − ε)n; see [17]. Dillencourt et al [26] studied the geometric thickness of K n , and proved that 5…”

Section: Open Problemsmentioning

confidence: 99%

Graph Treewidth and Geometric Thickness Parameters

Dujmović

Wood

2006

Graph Drawing

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Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth.Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215-221, 2001]. Analogous results are proved for outerthickness, arboricity, and star-arboricity.

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Partitions of complete geometric graphs into plane trees

Cited by 33 publications

References 12 publications

Packing 1-plane Hamiltonian cycles in complete geometric graphs

Packing 1-plane Hamiltonian cycles in complete geometric graphs

Order on Order Types

Graph Treewidth and Geometric Thickness Parameters

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