Forbidden Induced Subgraphs and the Price of Connectivity for Feedback Vertex Set

Abstract

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. For a graph class ${\cal G}$, the price of connectivity for feedback vertex set (poc-fvs) for ${\cal G}$ is defined as the maximum ratio cfvs(G)/fvs(G) over all connected graphs G in ${\cal G}$. It is known that the poc-fvs for general graphs is unbounded. We study the poc-fvs for graph classes defined by a finite family ${\cal H}$ of forbidden induced subgraphs. We characterize exactly those finite families ${\cal H}$ for which the poc-fvs for ${\cal H}$-free graphs is bounded by a constant. Prior to our work, such a result was only known for the case where $|{\cal H}|=1$.