Maintaining CMSO₂ Properties on Dynamic Structures with Bounded Feedback Vertex Number

Authors Konrad Majewski , Michał Pilipczuk , Marek Sokołowski

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Konrad Majewski
  • Institute of Informatics, University of Warsaw, Poland
Michał Pilipczuk
  • Institute of Informatics, University of Warsaw, Poland
Marek Sokołowski
  • Institute of Informatics, University of Warsaw, Poland

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Konrad Majewski, Michał Pilipczuk, and Marek Sokołowski. Maintaining CMSO₂ Properties on Dynamic Structures with Bounded Feedback Vertex Number. In 40th International Symposium on Theoretical Aspects of Computer Science (STACS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 254, pp. 46:1-46:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Let 𝜑 be a sentence of CMSO₂ (monadic second-order logic with quantification over edge subsets and counting modular predicates) over the signature of graphs. We present a dynamic data structure that for a given graph G that is updated by edge insertions and edge deletions, maintains whether 𝜑 is satisfied in G. The data structure is required to correctly report the outcome only when the feedback vertex number of G does not exceed a fixed constant k, otherwise it reports that the feedback vertex number is too large. With this assumption, we guarantee amortized update time O_{𝜑,k}(log n). By combining this result with a classic theorem of Erdős and Pósa, we give a fully dynamic data structure that maintains whether a graph contains a packing of k vertex-disjoint cycles with amortized update time O_k(log n). Our data structure also works in a larger generality of relational structures over binary signatures.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Mathematics of computing → Graph algorithms
  • feedback vertex set
  • CMSO₂ formula
  • data structure
  • dynamic graphs
  • fixed-parameter tractability


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