Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

Abstract A graph G is weakly $$\gamma $$ γ -closed if every induced subgraph of contains one vertex v such that for each non-neighbor u it holds \vert N(u)\cap N(v) <\gamma | N ( u ) ∩ v < . The weak closure (G)$$ G a graph, recently introduced by Fox et al. (SIAM J Comput 49(2):448–464, 2020), the smallest number -closed. This parameter never larger than degeneracy (plus one) and can be significantly smaller. Extending work (2020) on clique enumeration, we show several problems related to finding dense subgraphs, as enumeration bicliques s -plexes, are fixed-parameter tractable with respect Moreover, problem determining whether has at least k vertices belongs class $$\mathcal {G}$$ which closed under taking subgraphs admits kernel most k^2$$ k 2 vertices. Finally, provide algorithms Independent Dominating Set Clique when parameterized +k$$ + where solution size. Furthermore, does not admit polynomial constant standard assumptions.