Dag Realizations of Directed Degree Sequences

  • Annabell Berger
  • Matthias Müller-Hannemann
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6914)


We consider the following graph realization problem. Given a sequence \(S:={a_1 \choose b_1},\dots,{a_n \choose b_n}\) with \(a_i,b_i\in \mathbb{Z}_0^+\), does there exist an acyclic digraph (a dag, no parallel arcs allowed) G = (V,A) with labeled vertex set V: = {v 1,…,v n } such that for all v i  ∈ V indegree and outdegree of v i match exactly the given numbers a i and b i , respectively? The complexity status of this problem is open, while a generalization, the f-factor dag problem can be shown to be NP-complete. In this paper, we prove that an important class of sequences, the so-called opposed sequences, admit an O(n + m) realization algorithm, where n and \(m = \sum_{i=1}^n a_i = \sum_{i=1}^n b_i\) denote the number of vertices and arcs, respectively. For an opposed sequence it is possible to order all non-source and non-sink tuples such that a i  ≤ a i + 1 and b i  ≥ b i + 1. Our second contribution is a realization algorithm for general sequences which significantly improves upon a naive exponential-time algorithm. We also investigate a special and fast realization strategy “lexmax”, which fails in general, but succeeds in more than 97% of all sequences with 9 tuples.


Hamiltonian Path Topological Sorting Realization Algorithm Acyclic Digraph Prescribe Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Annabell Berger
    • 1
  • Matthias Müller-Hannemann
    • 1
  1. 1.Department of Computer ScienceMartin-Luther-Universität Halle-WittenbergGermany

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