Abstract
AbstractWe 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 O(w)·n, but it cannot be solved in time 2o(wlogk)·n O(1) 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 O(w)·n, but they cannot be solved in time 2o(wlogd)·n O(1) unless ETH fails.
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