On the complexity of the balanced vertex ordering problem

  • Authors:

  • Jan Kára;Jan Kratochvíl;David R. Wood

  • Affiliations:

  • Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic;Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic;Departament de Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Barcelona, Spain

  • Venue:
  • COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
  • Year:
  • 2005

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Abstract

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 Biedletal. [1]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but $\mathcal{NP}$-hard for graphs with maximum degree six. One of our main results is closing the gap in these results, by proving $\mathcal{NP}$-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains $\mathcal{NP}$-hard for planar graphs with maximum degree six and for 5-regular graphs. On the other hand we present a polynomial time algorithm that determines whether there 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.