Parameterized Complexity of the Sparsest k-Subgraph Problem in Chordal Graphs

In this paper we tackle the Sparsest k-Subgraph problem which consists in finding a subset of k vertices in a graph which induces the minimum number of edges. The Sparsest k-Subgraph problem is a natural generalization of the Independent Set problem, and thus is NP-hard (and even W [1]-hard) in general graphs. In this paper we investigate the parameterized complexity of both Sparsest k-Subgraph and Densest k-Subgraph in chordal graphs. We first provide simple proofs that Densest k-Subgraph in chordal graphs is FPT and does not admit a polynomial kernel unless NP ⊆ coNP/poly (both parameterized by k). More involved proofs will ensure the same behavior for Sparsest k-Subgraph in the same graph class. We lastly provide an FPT algorithm in interval graphs for Sparsest kSubgraph, but parameterized by the number of edges of the solution (a stronger parameterization than by k).