### 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.

Original language | English |
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Pages (from-to) | 27-48 |

Number of pages | 22 |

Journal | Discrete Applied Mathematics |

Volume | 148 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2005 |

Externally published | Yes |

## Cite this

Biedl, T., Chan, T., Ganjali, Y., Hajiaghayi, M. T., & Wood, D. R. (2005). Balanced vertex-orderings of graphs.

*Discrete Applied Mathematics*,*148*(1), 27-48. https://doi.org/10.1016/j.dam.2004.12.001