Nordhaus-Gaddum inequalities for the number of 1-nearly independent vertex subsets
Eric O. D. Andriantiana, Zekhaya B. Shozi
TL;DR
This work studies Nordhaus-Gaddum type inequalities for the count of 1-nearly independent vertex subsets, denoted $\sigma_1(G)$. It proves a tight lower bound $\sigma_1(G)+\sigma_1(\overline{G})\ge \frac{n(n-1)}{2}$ with equality only for complete or edgeless graphs, and identifies the unique tree minimizer as the star $K_{1,n-1}$. It then derives a tight upper bound for all graphs with $n\ge 6$, $\sigma_1(G)+\sigma_1(\overline{G}) \le \frac{27}{64}\,2^{n} + \frac{1}{2}(n+2)(n-3)$, with equality when $G$ or $\overline{G}$ is isomorphic to $3K_2 \cup \overline{K_{n-6}}$ (or the related graph $K_{n-6} \vee G_{6,4}$). The paper builds on a recursive computation for $\sigma_1(G)$ and the notion of good graphs to characterize extremal structures, linking to prior results on $\sigma_1$ and expanding the understanding of how this invariant behaves under complementation.
Abstract
For a graph $G$, a vertex subset is called \emph{$1$-nearly independent} if the subgraph it induces contains exactly one edge. Let $σ_1(G)$ denote the number of such subsets in $G$. In this paper, we study Nordhaus-Gaddum type inequalities for $σ_1$, that is, bounds on the sum $σ_1(G)+σ_1(\overline{G})$, where $\overline{G}$ denotes the complement of $G$. We establish that, for any $n$-vertex graph $G$, we have $σ_1(G)+σ_1(\overline{G})\geq n(n-1)/2,$ 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 $σ_1(T)+σ_1(\overline{T})$. Finally, we prove that for all graphs of order $n \ge 6$, \[ σ_1(G)+σ_1(\overline{G}) \le \frac{27}{64}\,2^{n} + \frac{1}{2}(n+2)(n-3), \] with equality if and only if $G$ or $\overline{G}$ is isomorphic to $3K_2 \cup \overline{K_{n-6}}$.