  A vertex whose removal in a graph $G$ increases the number of components of
$G$ is called a cut vertex. For all $n,c$, we determine the maximum number of
connected induced subgraphs in a connected graph with order $n$ and $c$ cut
vertices, and also characterise those graphs attaining the bound. Moreover, we
show that the cycle has the smallest number of connected induced subgraphs
among all cut vertex-free connected graphs. The general case $c>0$ remains an
open task. We also characterise the extremal graph structures given both order
and number of pendant vertices, and establish the corresponding formulas for
the number of connected induced subgraphs. The `minimal' graph in this case is
a tree, thus coincides with the structure that was given by Li and
Wang~[Further analysis on the total number of subtrees of trees.
\emph{Electron. J. Comb.} 19(4), #P48, 2012].
