The $1$-nearly vertex independence number of a graph

TL;DR

This paper introduces and analyzes the concept of the 1-nearly vertex independence number in graphs, providing explicit formulas, bounds, and characterizations of extremal graphs.

Contribution

It defines the 1-nearly vertex independence number, derives explicit formulas, establishes tight bounds, and characterizes extremal graphs achieving these bounds.

Findings

Explicit formulas for  in certain cases

Tight lower and upper bounds for  in graphs of order n

Complete characterization of extremal graphs

Abstract

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. A set $I_0(G) \subseteq V(G)$ is a vertex independent set if no two vertices in $I_0(G)$ are adjacent in $G$. We study $\alpha_1(G)$, which is the maximum cardinality of a set $I_1(G) \subseteq V(G)$ that contains exactly one pair of adjacent vertices of $G$. We call $I_1(G)$ a $1$-nearly vertex independent set of $G$ and $\alpha_1(G)$ a $1$-nearly vertex independence number of $G$. We provide some cases of explicit formulas for $\alpha_1$. Furthermore, we prove a tight lower (resp. upper) bound on $\alpha_1$ for graphs of order $n$. The extremal graphs that achieve equality on each bound are fully characterised.