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

TL;DR

This paper establishes bounds on the sum of the number of 1-nearly independent vertex subsets in a graph and its complement, characterizing extremal graphs and providing inequalities for all graphs and trees.

Contribution

It introduces Nordhaus-Gaddum inequalities for the count of 1-nearly independent subsets, identifying extremal graphs and deriving bounds for general graphs and trees.

Findings

Lower bound of n(n-1)/2 for complete or edgeless graphs.

Star graph minimizes the sum among all trees.

Upper bound achieved by specific disjoint unions of edges and independent sets.

Abstract

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 $\sigma_1(G)+\sigma_1(\overline{G})$, where $\overline{G}$ denotes the complement of $G$. We establish that, for any $n$-vertex graph $G$, we have $\sigma_1(G)+\sigma_1(\overline{G})\geq n(n-1)/2,$ 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 $\sigma_1(T)+\sigma_1(\overline{T})$. Finally, we prove that for all graphs of order $n \ge 6$, \[ \sigma_1(G)+\sigma_1(\overline{G}) \le \frac{27}{64}\,2^{n} + \frac{1}{2}(n+2)(n-3), \] with equality if and only if $G$ or $\overline{G}$ is isomorphic to $3K_2 \cup \overline{K_{n-6}}$.