  Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges
$E(G)$. For $k\ge 0$ an integer, a subset $I_k$ of $V(G)$ is called a
$k$-nearly independent vertex subset of $G$ if $I_k$ induces a subgraph of size
$k$ in $G$. The number of such subsets in $G$ is denoted by $\sigma_k(G)$. In
this paper we continue the study of $\sigma_1$. In particular, we prove the
lower bound on $\sigma_1$ for a connected graph that contains a cycle and also
characterise the two extremal graphs. This improves the result obtained in [E.
O. D. Andriantiana and Z. B. Shozi. The number of 1-nearly independent vertex
subsets. \textit{Quaestiones Mathematicae}, accepted].
