eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-12-04
20:1
20:17
10.4230/LIPIcs.ISAAC.2020.20
article
Computing Dense and Sparse Subgraphs of Weakly Closed Graphs
Koana, Tomohiro
1
https://orcid.org/0000-0002-8684-0611
Komusiewicz, Christian
2
https://orcid.org/0000-0003-0829-7032
Sommer, Frank
2
https://orcid.org/0000-0003-4034-525X
Technische Universität Berlin, Algorithmics and Computational Complexity, Germany
Philipps-Universität Marburg, Fachbereich Mathematik und Informatik, Germany
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol181-isaac2020/LIPIcs.ISAAC.2020.20/LIPIcs.ISAAC.2020.20.pdf
Fixed-parameter tractability
c-closure
degeneracy
clique relaxations
bicliques
dominating set