Nordhaus-Gaddum inequalities for the number of 1-nearly independent vertex subsets

For a graph $G$, a vertex subset is called \emph{$1$-nearly independent} if the subgraph it induces contains exactly one edge. Let ${\sigma }_1(G)$ denote the number of such subsets in $G$. In this paper, we study Nordhaus-Gaddum type inequalities for ${\sigma }_1$, that is, bounds on the sum ${\sigma }_1(G)+{\sigma }_1(\bar{G})$, where $\bar{G}$ denotes the complement of $G$. We establish that, for any $n$-vertex graph $G$, we have ${\sigma }_1(G)+{\sigma }_1(\bar{G})\ge n(n-1)\mathrm{/}2,$ with equality if and only if $G$ is either complete or edgeless. We further obtain that among all trees of order $n$, the star $K_{1,n-1}$ uniquely minimises ${\sigma }_1(T)+{\sigma }_1(\bar{T})$. Finally, we prove that for all graphs of order $n\ge 6$, $${\sigma }_1(G)+{\sigma }_1(\bar{G})\le \frac{27}{64}{2}^n+\frac{1}{2}(n+2)(n-3),$$ with equality if and only if $G$ or $\bar{G}$ is isomorphic to $3K_2\cup \overline{K_{n-6}}$.