An improved lower bound on the number of $1$-nearly independent vertex subsets

Let $G=(V(G),E(G))$ be a graph with set of vertices $V(G)$ and set of edges $E(G)$. For k≥ 0 an integer, a subset Iₖ of $V(G)$ is called a k-nearly independent vertex subset of G if Iₖ induces a subgraph of size k in G. The number of such subsets in G is denoted by σₖ(G). In this paper we continue the study of σ₁. In particular, we prove the lower bound on σ₁ 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. Quaestiones Mathematicae, accepted].