  We study the Feedback Vertex Set and the Vertex Cover problem in a natural
variant of the classical online model that allows for delayed decisions and
reservations. Both problems can be characterized by an obstruction set of
subgraphs that the online graph needs to avoid. In the case of the Vertex Cover
problem, the obstruction set consists of an edge (i.e., the graph of two
adjacent vertices), while for the Feedback Vertex Set problem, the obstruction
set contains all cycles.
  In the delayed-decision model, an algorithm needs to maintain a valid partial
solution after every request, thus allowing it to postpone decisions until the
current partial solution is no longer valid for the current request.
  The reservation model grants an online algorithm the new and additional
option to pay a so-called reservation cost for any given element in order to
delay the decision of adding or rejecting it until the end of the instance.
  For the Feedback Vertex Set problem, we first analyze the variant with only
delayed decisions, proving a lower bound of $4$ and an upper bound of $5$ on
the competitive ratio. Then we look at the variant with both delayed decisions
and reservation. We show that given bounds on the competitive ratio of a
problem with delayed decisions impliy lower and upper bounds for the same
problem when adding the option of reservations. This observation allows us to
give a lower bound of $\min{\{1+3\alpha,4\}}$ and an upper bound of
$\min{\{1+5\alpha,5\}}$ for the Feedback Vertex Set problem. Finally, we show
that the online Vertex Cover problem, when both delayed decisions and
reservations are allowed, is $\min{\{1+2\alpha, 2\}}$-competitive, where
$\alpha \in \mathbb{R}_{\geq 0}$ is the reservation cost per reserved vertex.
