## Abstract

We call a topological ordering of a weighted directed acyclic graph non-negative if the sum of weights on the vertices in any prefix of the ordering is non-negative. We investigate two processes for constructing non-negative topological orderings of weighted directed acyclic graphs. The first process is called a mark sequence and the second is a generalization called a mark-unmark sequence. We answer a question of Erickson by showing that every non-negative topological ordering that can be realized by a mark-unmark sequence can also be realized by a mark sequence. We also investigate the question of whether a given weighted directed acyclic graph has a non-negative topological ordering. We show that even in the simple case when every vertex is a source or a sink the question is NP-complete.

Original language | English |
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Pages (from-to) | 564-568 |

Number of pages | 5 |

Journal | Information Processing Letters |

Volume | 116 |

Issue number | 9 |

DOIs | |

State | Published - Sep 1 2016 |

## Keywords

- Directed acyclic graph
- Graph Algorithms
- Topological ordering