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 \(I_0(G) \subseteq V(G)\) is a vertex independent set if no two vertices in \(I_0(G)\) are adjacent in \(G\). We study \(\alpha_1(G)\), which is the maximum cardinality of a set \(I_1(G) \subseteq V(G)\) that contains exactly one pair of adjacent vertices of \(G\). We call \(I_1(G)\) a \(1\)-nearly vertex independent set of \(G\) and \(\alpha_1(G)\) a \(1\)-nearly vertex independence number of \(G\). We provide some cases of explicit formulas for \(\alpha_1\). Furthermore, we prove a tight lower (resp. upper) bound on \(\alpha_1\) for graphs of order \(n\). The extremal graphs that achieve equality on each bound are fully characterised.