Reconfiguration of Graphs with Connectivity Constraints

  • Nicolas BousquetEmail author
  • Arnaud Mary
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11312)


A graph G realizes the degree sequence S if the degrees of its vertices is S. Hakimi [5] gave a necessary and sufficient condition to guarantee that there exists a connected multigraph realizing S. Taylor [13] later proved that any connected multigraph can be transformed into any other via a sequence of flips (maintaining connectivity at any step). A flip consists in replacing two edges ab and cd by the diagonals ac and bd. In this paper, we study a generalization of this problem. A set of subsets of vertices \(\mathcal {CC}\) is nested if for every \(C,C' \in \mathcal {CC}\) either \(C \cap C' = \emptyset \) or one is included in the other. We are interested in multigraphs realizing a degree sequence S and such that all the sets of a nested collection \(\mathcal {CC}\) induce connected subgraphs. Such constraints naturally appear in tandem mass spectrometry.

We show that it is possible to decide in polynomial if there exists a graph realizing S where all the sets in \(\mathcal {CC}\) induce connected subgraphs. Moreover, we prove that all such graphs can be obtained via a sequence of flips such that all the intermediate graphs also realize S and where all the sets of \(\mathcal {CC}\) induce connected subgraphs. Our proof is algorithmic and provides a polynomial time approximation algorithm on the shortest sequence of flips between two graphs whose ratio depends on the depth of the nested partition.



The authors want to thank the anonymous reviewers of WAOA for their careful reading of the paper which permits to significantly improve its quality.


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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.CNRS, G-SCOP, Grenoble-INPUniv. Grenoble-AlpesGrenobleFrance
  2. 2.LBBE, Université Claude Bernard Lyon 1LyonFrance

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