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Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

Authors Tomohiro Koana , Christian Komusiewicz , Frank Sommer

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Author Details

Tomohiro Koana
  • Technische Universität Berlin, Algorithmics and Computational Complexity, Germany
Christian Komusiewicz
  • Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany
Frank Sommer
  • Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany

Funding

  • Koana, Tomohiro: Supported by the Deutsche Forschungsgemeinschaft (DFG), project FPTinP, NI 369/19.
  • Sommer, Frank: Supported by the Deutsche Forschungsgemeinschaft (DFG), project MAGZ, KO 3669/4-1.

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Tomohiro Koana, Christian Komusiewicz, and Frank Sommer. Computing Dense and Sparse Subgraphs of Weakly Closed Graphs. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ISAAC.2020.20

Abstract

A graph G is weakly γ-closed if every induced subgraph of G contains one vertex v such that for each non-neighbor u of v it holds that |N(u)∩ N(v)| < γ. The weak closure γ(G) of a graph, recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number such that G is weakly γ-closed. This graph parameter is never larger than the degeneracy (plus one) and can be significantly smaller. Extending the work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that several problems related to finding dense subgraphs, such as the enumeration of bicliques and s-plexes, are fixed-parameter tractable with respect to γ(G). Moreover, we show that the problem of determining whether a weakly γ-closed graph G has a subgraph on at least k vertices that belongs to a graph class 𝒢 which is closed under taking subgraphs admits a kernel with at most γ k² vertices. Finally, we provide fixed-parameter algorithms for Independent Dominating Set and Dominating Clique when parameterized by γ+k where k is the solution size.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • Fixed-parameter tractability
  • c-closure
  • degeneracy
  • clique relaxations
  • bicliques
  • dominating set

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