  Cut vertices are often used as a measure of nodes' importance within a
network. They are those nodes whose failure disconnects a graph. Let N(G) be
the number of connected induced subgraphs of a graph $G$. In this work, we
investigate the maximum of N(G) where $G$ is a unicyclic graph with $n$ nodes
of which $c$ are cut vertices. For all valid $n,c$, we give a full description
of those maximal (that maximise N(.)) unicyclic graphs. It is found that there
are generally two maximal unicyclic graphs. For infinitely many values of
$n,c$, however, there is a unique maximal unicyclic graph with $n$ nodes and
$c$ cut vertices. In particular, the well-known negative correlation between
the number of connected induced subgraphs of trees and the Wiener index (sum of
distances) fails for unicyclic graphs with $n$ nodes and $c$ cut vertices: for
instance, the maximal unicyclic graph with $n=3,4\mod 5$ nodes and $c=n-5>3$
cut vertices is different from the unique graph that was shown by Tan et
al.~[{\em The Wiener index of unicyclic graphs given number of pendant vertices
or cut vertices}. J. Appl. Math. Comput., 55:1--24, 2017] to minimise the
Wiener index. Our main characterisation of maximal unicyclic graphs with
respect to the number of connected induced subgraphs also applies to unicyclic
graphs with $n$ nodes, $c$ cut vertices and girth at most $g>3$, since it is
shown that the girth of every maximal graph with $n$ nodes and $c$ cut vertices
cannot exceed $4$.
