  The following optimal stopping problem is considered. The vertices of a graph
$G$ are revealed one by one, in a random order, to a selector. He aims to stop
this process at a time $t$ that maximizes the expected number of connected
components in the graph $\tilde{G}_t$, induced by the currently revealed
vertices. The selector knows $G$ in advance, but different versions of the game
are considered depending on the information that he gets about $\tilde{G}_t$.
We show that when $G$ has $N$ vertices and maximum degree of order
$o(\sqrt{N})$, then the number of components of $\tilde{G}_t$ is concentrated
around its mean, which implies that playing the optimal strategy the selector
does not benefit much by receiving more information about $\tilde{G}_t$.
Results of similar nature were previously obtained by M. Laso\'n for the case
where $G$ is a $k$-tree (for constant $k$). We also consider the particular
cases where $G$ is a square, triangular or hexagonal lattice, showing that an
optimal selector gains $cN$ components and we compute $c$ with an error less
than $0.005$ in each case.
