Relating broadcast independence and independence

TL;DR

This paper establishes a tight bound relating broadcast independence number to the classical independence number in connected graphs, proving that the broadcast independence number is at most four times the independence number and characterizing extremal cases.

Contribution

The paper introduces a new inequality linking broadcast independence and independence numbers, providing a tight bound and characterizing extremal graphs.

Findings

Proved that (G)  4(G) for connected graphs.

Established the inequality as tight with characterization of extremal graphs.

Enhanced understanding of broadcast independence in relation to classical independence.

Abstract

An independent broadcast on a connected graph $G$ is a function $f:V(G)\to \mathbb{N}_0$ such that, for every vertex $x$ of $G$, the value $f(x)$ is at most the eccentricity of $x$ in $G$, and $f(x)>0$ implies that $f(y)=0$ for every vertex $y$ of $G$ within distance at most $f(x)$ from $x$. The broadcast independence number $\alpha_b(G)$ of $G$ is the largest weight $\sum\limits_{x\in V(G)}f(x)$ of an independent broadcast $f$ on $G$. Clearly, $\alpha_b(G)$ is at least the independence number $\alpha(G)$ for every connected graph $G$. Our main result implies $\alpha_b(G)\leq 4\alpha(G)$. We prove a tight inequality and characterize all extremal graphs.