The Price of Connectivity for Feedback Vertex Set

Let fvs(G) and cfvs(G) denote the cardinalities of a minimum feedback vertex set and a minimum connected feedback vertex set of a graph G, respectively. In general graphs, the ratio cfvs(G)/fvs(G) can be arbitrarily large. We study the interdependence between fvs(G) and cfvs(G) in graph classes defined by excluding one induced subgraph H. We show that the ratio cfvs(G)/fvs(G) is bounded by a constant for every connected H-free graph G if and only if H is a linear forest. We also determine exactly those graphs H for which there exists a constant cH such that cfvs(G) ≤ fvs(G) + cH for every connected H-free graph G, as well as exactly those graphs H for which we can take cH = 0.