Structure of a Maximal Total Independent Set

TL;DR

This paper explores the relationships between different types of independence numbers in graphs and characterizes the structures of maximum total independent sets.

Contribution

It establishes a new relation connecting total independence number with other graph invariants and describes the possible structures of maximum total independent sets.

Findings

Derived a relation between total independence number and other graph parameters

Characterized the structures of maximum total independent sets

Provides theoretical insights into graph independence properties

Abstract

Let $G$ be a simple, connected and finite graph with order $n$. Denote the independence number, edge independence number and total independence number by $\alpha(G), \alpha'(G)$ and $\alpha"(G)$ respectively. This paper establishes a relation between $\alpha"(G)$ with $\alpha(G)$, $\alpha'(G)$ and $n$. It also describes the possible structures of a total independent set of a given size.