The $1$-nearly vertex independence number of a graph

Let G be a graph with vertex set $V(G)$ and edge set $E(G)$. A set I₀(G) ⊆ V(G) is a vertex independent set if no two vertices in I₀(G) are adjacent in G. We study α₁(G), which is the maximum cardinality of a set I₁(G) ⊆ V(G) that contains exactly one pair of adjacent vertices of G. We call I₁(G) a $1$-nearly vertex independent set of G and α₁(G) a $1$-nearly vertex independence number of G. We provide some cases of explicit formulas for α₁. Furthermore, we prove a tight lower (resp. upper) bound on α₁ for graphs of order n. The extremal graphs that achieve equality on each bound are fully characterised.