On the Broadcast Independence Number of Caterpillars

TL;DR

This paper investigates the broadcast independence number of caterpillar graphs, providing an explicit formula for a specific subclass where no two adjacent vertices have degree 2, advancing understanding of broadcast properties in these trees.

Contribution

It introduces an explicit formula for the broadcast independence number of caterpillars without adjacent degree-2 vertices, a novel result in graph broadcast theory.

Findings

Derived an explicit formula for a subclass of caterpillars

Characterized the broadcast independence number for these trees

Enhanced understanding of broadcast functions in tree graphs

Abstract

Let $G$ be a simple undirected graph.A broadcast on $G$ isa function $f : V(G)\rightarrow\mathbb{N}$ such that $f(v)\le e\_G(v)$ holds for every vertex $v$ of $G$, where $e\_G(v)$ denotes the eccentricity of $v$ in $G$, that is, the maximum distance from $v$ to any other vertex of $G$.The cost of $f$ is the value ${\rm cost}(f)=\sum\_{v\in V(G)}f(v)$.A broadcast $f$ on $G$ is independent if for every two distinct vertices $u$ and $v$ in $G$, $d\_G(u,v)>\max\{f(u),f(v)\}$,where $d\_G(u,v)$ denotes the distance between $u$ and $v$ in $G$.The broadcast independence number of $G$ is then defined as the maximum cost of an independent broadcast on $G$. In this paper, we study independent broadcasts of caterpillars and give an explicit formula for the broadcast independence number of caterpillars having no pair of adjacent vertices with degree 2.