The Price of Connectivity for Feedback Vertex Set

TL;DR

This paper investigates the ratio between the size of a minimum connected feedback vertex set and a minimum feedback vertex set in graphs, characterizing graph classes where this ratio is bounded or can be tightly estimated.

Contribution

It characterizes finite families of forbidden subgraphs for which the price of connectivity for feedback vertex set is bounded, and precisely identifies graphs where the ratio is close to 1.

Findings

Identifies when the ratio is bounded by a constant for H-free graphs.

Determines graphs H for which cfvs(G) is at most fvs(G) plus a constant.

Classifies graphs H where cfvs(G) equals fvs(G) for all connected H-free graphs.

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. The price of connectivity for feedback vertex set (poc-fvs) for a class of graphs ${\cal G}$ is defined as the maximum ratio $\mbox{cfvs}(G)/\mbox{fvs}(G)$ over all connected graphs $G\in {\cal G}$. 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 upper bounded by a constant. Additionally, for the case where $|{\cal H}|=1$, we determine exactly those graphs $H$ for which there exists a constant $c_H$ such that $\mbox{cfvs}(G)\leq \mbox{fvs}(G) + c_H$ for every connected $H$-free graph $G$, as well as exactly those graphs $H$ for which we can take $c_H=0$.