Tight Complexity Bounds for FPT Subgraph Problems Parameterized by Clique-Width

We give tight algorithmic lower and upper bounds for some double-parameterized subgraph problems when the clique-width of the input graph is one of the parameters. Let G be an arbitrary input graph on n vertices with clique-width at most w. We prove the following results. – The Dense (Sparse) k-Subgraph problem, which asks whether there exists an induced subgraph of G with k vertices and at least q edges (at most q edges, respectively), can be solved in time k ·n, but it cannot be solved in time 2 log k) ·n unless the Exponential Time Hypothesis (ETH) fails. – The d-Regular Induced Subgraph problem, which asks whether there exists a d-regular induced subgraph of G, and the Minimum Subgraph of Minimum Degree at least d problem, which asks whether there exists a subgraph of G with k vertices and minimum degree at least d, can be solved in time d · n, but they cannot be solved in time 2 log d) · n unless ETH fails.