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, $av_1(G)$, in $n$-vertex graphs and in $n$-vertex trees. It develops the $l$-nearly independent framework with $I_l(G;x)$, $\sigma_l(G)$ and $S_l(G)$, and derives edge-decomposition and vertex-removal recurrences to obtain sharp extremal bounds. The main results show that every non-edgeless graph satisfies $av_1(G)\ge 2$ with equality characterizing a family $\mathcal{H}$ of graphs, and that $av_1(G)\le n/2+1$ with equality only for $G_n=K_2\cup(n-2)K_1$ when $n\ge 6$; among trees, the star minimizes $av_1$ while a constructed family $R_n$ yields $av_1(R_n)=n/2+\delta_n$ with $0<\delta_n<1/2$, showing asymptotic sharpness and suggesting $R_n$ attains the maximum. The results tie $av_1$ tightly to $av_0$ via edge-based decompositions and provide asymptotically sharp bounds, advancing the extremal theory of nearly independent sets.

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.

Theorems & Definitions (25)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• Proposition 4
• proof
• Lemma 5
• Lemma 6: andriantiana2018
• Lemma 7
• proof