Compatible spanning trees

García, Alfredo; Huemer, Clemens; Hurtado, Ferrán; Tejel, Javier

doi:10.1016/j.comgeo.2013.12.009

Cited by 6 publications

(1 citation statement)

References 27 publications

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“…Two simultaneous exchanges suffice if there exists a tree T 3 = (S, E 3 ) such that E 1 ∩ E 3 = ∅. This strategy does not work directly for simultaneous compatible exchanges: García et al [20] constructed a tree T 1 ∈ T (S) such that any compatible T 2 ∈ T (S) has at least (n − 2)/5 edges in common with T 1 .…”

Section: General Positionmentioning

confidence: 99%

LATIN 2018: Theoretical Informatics

2018

Lecture Notes in Computer Science

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The operation of transforming one spanning tree into another by replacing an edge has been considered widely, both for general and planar straight-line graphs. For the latter, several variants have been studied (e.g., edge slides and edge rotations). In a transition graph on the set T (S) of noncrossing straight-line spanning trees on a finite point set S in the plane, two spanning trees are connected by an edge if one can be transformed into the other by such an operation. We study bounds on the diameter of these graphs, and consider the various operations both on general point sets and sets in convex position. In addition, we address problem variants where operations may be performed simultaneously or the edges are labeled. We prove new lower and upper bounds for the diameters of the corresponding transition graphs and pose open problems.

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Section: General Positionmentioning

confidence: 99%

LATIN 2018: Theoretical Informatics

2018

Lecture Notes in Computer Science

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The Mathematics of Ferran Hurtado: A Brief Survey

Urrutia

2016

Lecture Notes in Computer Science

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Plane augmentation of plane graphs to meet parity constraints

Catana

García

Tejel

et al. 2020

Applied Mathematics and Computation

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A plane topological graph G = (V, E) is a graph drawn in the plane whose vertices are points in the plane and whose edges are simple curves that do not intersect, except at their endpoints. Given a plane topological graph G = (V, E) and a set CG of parity constraints, in which every vertex has assigned a parity constraint on its degree, either even or odd, we say that G is topologically augmentable to meet CG if there exits a plane topological graph H on the same set of vertices, such that G and H are edge-disjoint and their union is a plane topological graph that meets all parity constraints. In this paper, we prove that the problem of deciding if a plane topological graph is topologically augmentable to meet parity constraints is N P-complete, even if the set of vertices that must change their parities is V or the set of vertices with odd degree. In particular, deciding if a plane topological graph can be augmented to a Eulerian plane topological graph is N P-complete. Analogous complexity results are obtained, when the augmentation must be done by a plane topological perfect matching between the vertices not meeting their parities. We extend these hardness results to planar graphs, when the augmented graph must be planar, and to plane geometric graphs (plane topological graphs whose edges are straight-line segments). In addition, when it is required that the augmentation is made by a plane geometric perfect matching between the vertices not meeting their parities, we also prove that this augmentation problem is N P-complete for plane geometric trees and paths. For the particular family of maximal outerplane graphs, we characterize maximal outerplane graphs that are topological augmentable to satisfy a set of parity constraints. We also provide a polynomial time algorithm that decides if a maximal outerplane graph is topologically augmentable to meet parity constraints, and if so, produces a set of edges with minimum cardinality.

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Compatible spanning trees

Cited by 6 publications

References 27 publications

LATIN 2018: Theoretical Informatics

LATIN 2018: Theoretical Informatics

The Mathematics of Ferran Hurtado: A Brief Survey

Plane augmentation of plane graphs to meet parity constraints

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