Tight Running Time Lower Bounds for Vertex Deletion Problems

For a graph class Π, the Π-V ertex D eletion problem has as input an undirected graph G = ( V , E ) and an integer k and asks whether there is a set of at most k vertices that can be deleted from G such that the resulting graph is a member of Π. By a classic result of Lewis and Yannakakis [17], Π-V ertex D eletion is NP-hard for all hereditary properties Π. We adapt the original NP-hardness construction to show that under the exponential time hypothesis (ETH), tight complexity results can be obtained. We show that Π-V ertex D eletion does not admit a 2 o ( n ) -time algorithm where n is the number of vertices in G . We also obtain a dichotomy for running time bounds that include the number m of edges in the input graph. On the one hand, if Π contains all edgeless graphs, then there is no 2 o ( n+m ) -time algorithm for Π-V ertex D eletion . On the other hand, if there is a fixed edgeless graph that is not contained in Π and containment in Π can be determined in 2 O ( n ) time or 2 o ( m ) time, then Π-V ertex D eletion can be solved in 2 O (√ m ) + O ( n ) or 2 o ( m ) + O ( n ) time, respectively. We also consider restrictions on the domain of the input graph G . For example, we obtain that Π-V ertex D eletion cannot be solved in 2 o (√ n ) time if G is planar and Π is hereditary and contains and excludes infinitely many planar graphs. Finally, we provide similar results for the problem variant where the deleted vertex set has to induce a connected graph.