Advertisement

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)

Abstract

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.

Notes

Acknowledgments

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

References

  1. 1.
    Bereg, S., Ito, H.: Transforming graphs with the same graphic sequence. J. Inf. Process. 25, 627–633 (2017).  https://doi.org/10.2197/ipsjjip.25.627CrossRefGoogle Scholar
  2. 2.
    Bonamy, M., Bousquet, N.: Recoloring graphs via tree decompositions. Eur. J. Comb. 69, 200–213 (2018).  https://doi.org/10.1016/j.ejc.2017.10.010MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cereceda, L., van den Heuvel, J., Johnson, M.: Finding paths between 3-colorings. J. Graph Theory 67(1), 69–82 (2011).  https://doi.org/10.1002/jgt.20514MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cooper, C., Dyer, M., Greenhill, C.: Sampling regular graphs and a peer-to-peer network. Comb. Probab. Comput. 16(4), 557–593 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hakimi, S.L.: On realizability of a set of integers as degrees of the vertices of a linear graph. I. J. Soc. Ind. Appl. Math. 10(3), 496–506 (1962). http://www.jstor.org/stable/2098746MathSciNetCrossRefGoogle Scholar
  6. 6.
    Hakimi, S.L.: On realizability of a set of integers as degrees of the vertices of a linear graph II. Uniqueness. J. Soc. Ind. Appl. Math. 11(1), 135–147 (1963). http://www.jstor.org/stable/2098770MathSciNetCrossRefGoogle Scholar
  7. 7.
    Havel, V.: A remark on the existence of finite graphs. Casopis Pest. Mat. 80, 477–480 (1955)zbMATHGoogle Scholar
  8. 8.
    Hearn, R.A., Demaine, E.: PSPACE-completeness of sliding-block puzzles and other problems through the nondeterministic constraint logic model of computation. Theor. Comput. Sci. 343(1–2), 72–96 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    van den Heuvel, J.: The Complexity of change. In: Blackburn, S.R., Gerke, S., Wildon, M. (eds.) Part of London Mathematical Society Lecture Note Series, p. 409 (2013)Google Scholar
  10. 10.
    Mohar, B., Salas, J.: On the non-ergodicity of the Swendsen-Wang-Koteckỳ algorithm on the Kagomé lattice. J. Stat. Mech. Theory Exp. 2010(05), P05016 (2010)CrossRefGoogle Scholar
  11. 11.
    Nishimura, N.: Introduction to reconfiguration (2017). PreprintGoogle Scholar
  12. 12.
    Senior, J.: Partitions and their representative graphs. Am. J. Math. 73(3), 663–689 (1951)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Taylor, R.: Contrained switchings in graphs. In: McAvaney, K.L. (ed.) Combinatorial Mathematics VIII. LNM, vol. 884, pp. 314–336. Springer, Heidelberg (1981).  https://doi.org/10.1007/BFb0091828CrossRefGoogle Scholar
  14. 14.
    Will, T.G.: Switching distance between graphs with the same degrees. SIAM J. Discrete Math. 12(3), 298–306 (1999).  https://doi.org/10.1137/S0895480197331156MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations