Jan Kára ; Jan Kratochvil ; David R. Wood - On the complexity of the balanced vertex ordering problem

dmtcs:383 - Discrete Mathematics & Theoretical Computer Science, January 1, 2007, Vol. 9 no. 1 - https://doi.org/10.46298/dmtcs.383
On the complexity of the balanced vertex ordering problemArticle

Authors: Jan Kára ORCID1; Jan Kratochvil ORCID1; David R. Wood ORCID2

  • 1 Department of Applied Mathematics [Prague]
  • 2 Departament de Matemàtica Aplicada II

We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedl et al. [Discrete Applied Math. 148:27―48, 2005]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but NP-hard for graphs with maximum degree six. One of our main results is to close the gap in these results, by proving NP-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains NP-hard for planar graphs with maximum degree four and for 5-regular graphs. On the other hand, we introduce a polynomial time algorithm that determines whetherthere is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an 'almost balanced' ordering.


Volume: Vol. 9 no. 1
Section: Graph and Algorithms
Published on: January 1, 2007
Imported on: March 26, 2015
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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