Maximizing the expected number of components in an online search of a graph

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.