Generalized Feedback Vertex Set Problems on Bounded-Treewidth Graphs: Chordality is the Key to Single-Exponential Parameterized Algorithms - Bonnet, Édouard; Brettell, Nick; Kwon, O-joung; Marx, Dániel (Algorithmica, 2019)
It has long been known that Feedback Vertex Set can be solved in time 2 O ( w log w ) n O ( 1 ) on n -vertex graphs of treewidth w , but it was only recently that this running time was improved to 2 O ( w ) n O ( 1 ) , that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class P of graphs, the Bounded P -Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d , whether G contains a set S of at most k vertices such that each block of G - S has at most d vertices and is in P . Assuming that P is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of d : if P consists only of chordal graphs, then the problem can be solved in time 2 O ( w d 2 ) n O ( 1 ) , if P contains a graph with an induced cycle of length ℓ ⩾ 4 , then the problem is not solvable in time 2 o ( w log w ) n O ( 1 ) even for fixed d = ℓ , unless the ETH fails. We also study a similar problem, called Bounded P - Component Vertex Deletion , where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if d is part of the input and P contains all chordal graphs, then it cannot be solved in time f ( w ) n o ( w ) for some function f , unless the ETH fails.