Balanced vertex-orderings of graphs

  • Authors:

  • Therese Biedl;Timothy Chan;Yashar Ganjali;Mohammad Taghi Hajiaghayi;David R. Wood

  • Affiliations:

  • School of Computer Science, University of Waterloo, Waterloo, Canada ON N2L 3G1;School of Computer Science, University of Waterloo, Waterloo, Canada ON N2L 3G1;Department of Electrical Engineering, Stanford University, Stanford, CA 94305, USA;Laboratory for Computer Science, Massachusetts Institute of Technology, Cambridge, MA 02139, USA;School of Computer Science, Carleton University, Ottawa, Canada ON K1S 5B6

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2005

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Abstract

In this paper we consider the problem of determining a balanced ordering of the vertices of a graph; that is, the neighbors of each vertex v are as evenly distributed to the left and right of v as possible. This problem, which has applications in graph drawing for example, is shown to be NP-hard, and remains NP-hard for bipartite simple graphs with maximum degree six. We then describe and analyze a number of methods for determining a balanced vertex-ordering, obtaining optimal orderings for directed acyclic graphs, trees, and graphs with maximum degree three. For undirected graphs, we obtain a 13/8-approximation algorithm. Finally we consider the problem of determining a balanced vertex-ordering of a bipartite graph with a fixed ordering of one bipartition. When only the imbalances of the fixed vertices count, this problem is shown to be NP-hard. On the other hand, we describe an optimal linear time algorithm when the final imbalances of all vertices count. We obtain a linear time algorithm to compute an optimal vertex-ordering of a bipartite graph with one bipartition of constant size.