  Let $vc(G)$, $fvs(G)$ and $oct(G)$, respectively, denote the size of a
minimum vertex cover, minimum feedback vertex set and minimum odd cycle
transversal in a graph $G$. One can ask, when looking for these sets in a
graph, how much bigger might they be if we require that they are independent;
that is, what is the price of independence? If $G$ has a vertex cover, feedback
vertex set or odd cycle transversal that is an independent set, then we let
$ivc(G)$, $ifvs(G)$ or $ioct(G)$, respectively, denote the minimum size of such
a set. Similar to a recent study on the price of connectivity (Hartinger et al.
EuJC 2016), we investigate for which graphs $H$ the values of $ivc(G)$,
$ifvs(G)$ and $ioct(G)$ are bounded in terms of $vc(G)$, $fvs(G)$ and $oct(G)$,
respectively, when the graph $G$ belongs to the class of $H$-free graphs. We
find complete classifications for vertex cover and feedback vertex set and an
almost complete classification for odd cycle transversal (subject to three
non-equivalent open cases). We also investigate for which graphs $H$ the values
of $ivc(G)$, $ifvs(G)$ and $ioct(G)$ are equal to $vc(G)$, $fvs(G)$ and
$oct(G)$, respectively, when the graph $G$ belongs to the class of $H$-free
graphs. We find a complete classification for vertex cover and almost complete
classifications for feedback vertex set (subject to one open case) and odd
cycle transversal (subject to three open cases).
