Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

Abstract

A graph G is weakly $$\gamma $$ γ -closed if every induced subgraph of G contains one vertex v such that for each non-neighbor u of v it holds that $$ \vert N(u)\cap N(v) \vert <\gamma $$ | N ( u ) ∩ N ( v ) | < γ . The weak closure $$\gamma (G)$$ γ ( G ) of a graph, recently introduced by Fox et al. (SIAM J Comput 49(2):448–464, 2020), is the smallest number such that G is weakly $$\gamma $$ 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. (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 $$\gamma (G)$$ γ ( G ) . Moreover, we show that the problem of determining whether a weakly $$\gamma $$ γ -closed graph G has a subgraph on at least k vertices that belongs to a graph class $$\mathcal {G}$$ G which is closed under taking subgraphs admits a kernel with at most $$\gamma k^2$$ γ k 2 vertices. Finally, we provide fixed-parameter algorithms for Independent Dominating Set and Dominating Clique when parameterized by $$\gamma +k$$ γ + k where k is the solution size. Furthermore, we show that Independent Dominating Set does not admit a polynomial kernel for constant $$\gamma $$ γ under standard assumptions.