On the average size of $1$-nearly independent vertex sets in graphs

TL;DR

This paper studies the average size of 1-nearly independent vertex sets in graphs and trees, identifying extremal structures and asymptotic bounds for these averages.

Contribution

It characterizes graphs and trees that maximize or minimize the average size of 1-nearly independent sets, providing asymptotic bounds and constructions.

Findings

Maximal average size is between n/2 and (n+1)/2 for large n.

Identifies extremal graphs and trees for the average size.

Constructs families of trees demonstrating the bounds are sharp.

Abstract

A $k$-nearly independent vertex subset of a graph $G$ is a set of vertices that induces a subgraph containing exactly $k$ edges. For $k = 0$, this coincides with the classical notion of independent subsets. This paper investigates the average size, $av_1(G)$ of the $1$-nearly independent vertex subsets of both graphs and trees of a given order $n$. Let $E_n$ denote the $n$-vertex edgeless graph, so that $av_1(E_n) = 0$. We determine all $n$-vertex graphs $G\neq E_n$ that minimize or maximize $av_1$. Similarly, we identify the trees of order $n$ that achieve the minimum value of $av_1$, and prove that the maximum value lies between $n/2$ and $(n+1)/2$ if $n>8$. Finally, we construct a family of $n$-vertex trees which shows that the bounds are asymptotically sharp.