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The K-discretization and K-incident graphs for discretizable Distance Geometry

  • Germano Abud
  • Jorge Alencar
  • Carlile Lavor
  • Leo Liberti
  • Antonio Mucherino
Original Paper

Abstract

The Distance Geometry Problem (DGP) is the problem of determining whether a realization for a simple weighted undirected graph \(G=(V,E,d)\) in a given Euclidean space exists so that the distances between pairs of realized vertices u, \(v \in V\) correspond to the weights \(d_{uv}\), for each \(\{u,v\} \in E\). We focus on a special class of DGP instances, referred to as the Discretizable DGP (DDGP), and we introduce the K-discretization and the K-incident graphs for the DDGP class. The K-discretization graph is independent on the vertex order that can be assigned to V, and can be useful for discovering whether one of such orders actually exists so that the DDGP assumptions are satisfied. The use of a given vertex order allows the definition of another important graph, the K-incident graph, which is potentially useful for performing pre-processing analysis on the solution set of DDGP instances.

Keywords

Distance geometry Vertex orders Discretization Combinatorial optimization 

Notes

Acknowledgements

We wish to thank the anonymous referees for the very fruitful comments. We are also grateful to the Brazilian research agencies FAPESP and CNPq for financial support.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.FAMATFederal University of UberlândiaUberlândiaBrazil
  2. 2.Federal Institute of the South of Minas GeraisBelo HorizonteBrazil
  3. 3.IMECCUniversity of CampinasCampinasBrazil
  4. 4.CNRS LIXÉcole PolytechniquePalaiseauFrance
  5. 5.IRISA and University of Rennes 1RennesFrance

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