Edge-disjoint Placement of Three Trees

Mahéo, Maryvonne; Saclé, Jean-François; Woźniak, Mariusz

doi:10.1006/eujc.1996.0047

Cited by 14 publications

(7 citation statements)

References 3 publications

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“…A star is a n-vertex tree where one vertex has degree n − 1. Further, Maheo et al [7] gave a characterization of which triples of non-star trees can be packed in K n . The planar packing problem is the variant of the packing problem in which the n-vertex graph G containing the given n-vertex graphs G 1 , .…”

Section: Introductionmentioning

confidence: 98%

Planar packing of trees and spider trees

Frati¹,

Geyer²,

Kaufmann³

2009

Information Processing Letters

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Section: Introductionmentioning

confidence: 98%

Planar packing of trees and spider trees

Frati¹,

Geyer²,

Kaufmann³

2009

Information Processing Letters

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“…Notice that, the hypothesis that the trees are not stars is necessary for the existence of the packing because each vertex must have degree at least one in each tree, which is not possible if a vertex is adjacent to every other vertex as it is the case for a star. Wang and Sauer [17] give sufficient conditions for the existence of a packing of three trees into K n , while Mahéo et al [12] characterize the triples of trees that admit such a packing. García et al [6] consider the planar packing problem, that is the case when the graph G is required to be planar.…”

Section: Introductionmentioning

confidence: 99%

Packing Trees into 1-Planar Graphs

Luca

Giacomo

Hong

et al. 2020

WALCOM: Algorithms and Computation

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We introduce and study the 1-planar packing problem: Given k graphs with n vertices G1, . . . , G k , find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each Gi is a tree and k = 3. We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1-planar packing, while two paths and a special type of caterpillar always have one. We then study 1-planar packings with few crossings and prove that three paths (resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with n ≥ 12 vertices admits a 1-planar packing, while such a packing does not exist if n ≤ 10.

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“…Teo and Yap [22] showed, extending an earlier result by Bollobás and Eldridge [2], that any two graphs of maximum degree at most n − 1 with a total of at most 2n − 2 edges pack into K n unless they are one of thirteen specified pairs of graphs. Maheo et al [17] characterized triples of trees that can be packed into K n .…”

Section: Introductionmentioning

confidence: 99%

The Planar Tree Packing Theorem

Geyer,

Hoffmann,

Kaufmann

et al. 2016

Preprint

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Packing graphs is a combinatorial problem where several given graphs are being mapped into a common host graph such that every edge is used at most once. In the planar tree packing problem we are given two trees T 1 and T 2 on n vertices and have to find a planar graph on n vertices that is the edge-disjoint union of T 1 and T 2 . A clear exception that must be made is the star which cannot be packed together with any other tree. But according to a conjecture of García et al. from 1997 this is the only exception, and all other pairs of trees admit a planar packing. Previous results addressed various special cases, such as a tree and a spider tree, a tree and a caterpillar, two trees of diameter four, two isomorphic trees, and trees of maximum degree three. Here we settle the conjecture in the affirmative and prove its general form, thus making it the planar tree packing theorem. The proof is constructive and provides a polynomial time algorithm to obtain a packing for two given nonstar trees.

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Edge-disjoint Placement of Three Trees

Cited by 14 publications

References 3 publications

Planar packing of trees and spider trees

Planar packing of trees and spider trees

Packing Trees into 1-Planar Graphs

The Planar Tree Packing Theorem

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