Computing Dense and Sparse Subgraphs of Weakly Closed Graphs
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.
Fixed-parameter tractability
c-closure
degeneracy
clique relaxations
bicliques
dominating set
Theory of computation~Parameterized complexity and exact algorithms
Theory of computation~Graph algorithms analysis
20:1-20:17
Regular Paper
A continously updated version of the paper is available at https://arxiv.org/abs/2007.05630.
Tomohiro
Koana
Tomohiro Koana
Technische Universität Berlin, Algorithmics and Computational Complexity, Germany
https://orcid.org/0000-0002-8684-0611
Supported by the Deutsche Forschungsgemeinschaft (DFG), project FPTinP, NI 369/19.
Christian
Komusiewicz
Christian Komusiewicz
Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany
https://orcid.org/0000-0003-0829-7032
Frank
Sommer
Frank Sommer
Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany
https://orcid.org/0000-0003-4034-525X
Supported by the Deutsche Forschungsgemeinschaft (DFG), project MAGZ, KO 3669/4-1.
10.4230/LIPIcs.ISAAC.2020.20
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Tomohiro Koana, Christian Komusiewicz, and Frank Sommer
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