pp 1–17 | Cite as

On Bubble Generators in Directed Graphs

  • V. Acuña
  • R. Grossi
  • G. F. Italiano
  • L. Lima
  • R. Rizzi
  • G. Sacomoto
  • M.-F. Sagot
  • B. SinaimeriEmail author


Bubbles are pairs of internally vertex-disjoint (st)-paths in a directed graph, which have many applications in the processing of DNA and RNA data. Listing and analysing all bubbles in a given graph is usually unfeasible in practice, due to the exponential number of bubbles present in real data graphs. In this paper, we propose a notion of bubble generator set, i.e., a polynomial-sized subset of bubbles from which all the other bubbles can be obtained through a suitable application of a specific symmetric difference operator. This set provides a compact representation of the bubble space of a graph. A bubble generator can be useful in practice, since some pertinent information about all the bubbles can be more conveniently extracted from this compact set. We provide a polynomial-time algorithm to decompose any bubble of a graph into the bubbles of such a generator in a tree-like fashion. Finally, we present two applications of the bubble generator on a real RNA-seq dataset.


Bubbles Bubble generator set Decomposition algorithm 



V. Acuña is supported by Fondecyt 1140631, PIA Fellowship AFB170001 and Center for Genome Regulation FONDAP 15090007. R. Grossi and G. F. Italiano are partially supported by MIUR, the Italian Ministry for Education, University and Research, under PRIN Project AHeAD (Efficient Algorithms for HArnessing Networked Data). Part of this work was done while G. F. Italiano was visiting Université de Lyon. L. Lima is supported by the Brazilian Ministry of Science, Technology and Innovation (in portuguese, Ministério da Ciência, Tecnologia e Inovação - MCTI) through the National Counsel of Technological and Scientific Development (in portuguese, Conselho Nacional de Desenvolvimento Científico e Tecnológico - CNPq), under the Science Without Borders (in portuguese, Ciências Sem Fronteiras) scholarship grant Process No. 203362/2014-4. B. Sinaimeri, L. Lima and M.-F. Sagot are partially funded by the French ANR project Aster (2016–2020), and together with V. Acuña, also by the Stic AmSud project MAIA (2016–2017). This work was performed using the computing facilities of the CC LBBE/PRABI.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Center for Mathematical ModelingUniversidad de Chile and UMI CNRS 2807SantiagoChile
  2. 2.Università di PisaPisaItaly
  3. 3.Erable, INRIAVilleurbanneFrance
  4. 4.LUISS UniversityRomeItaly
  5. 5.Università di VeronaVeronaItaly
  6. 6.Erable INRIA Rhône-Alpes; Université Lyon 1, CNRS, LBBE, UMR 5558VilleurbanneFrance
  7. 7.Università di Roma “Tor Vergata”RomeItaly

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