Algorithmic aspects of broadcast independence

TL;DR

This paper introduces an efficient algorithm for computing the broadcast independence number in trees, proves NP-hardness for planar graphs with degree four, and discusses approximation hardness for general graphs.

Contribution

It provides the first polynomial-time algorithm for trees and establishes complexity results for planar and general graphs related to broadcast independence.

Findings

Efficient algorithm for trees' broadcast independence number

NP-hardness for planar graphs with max degree four

Hardness of approximation for general graphs

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 $G$. We describe an efficient algorithm that determines the broadcast independence number of a given tree. Furthermore, we show NP-hardness of the broadcast independence number for planar graphs of maximum degree four, and hardness of approximation for general graphs. Our results solve problems posed by Dunbar, Erwin, Haynes, Hedetniemi, and Hedetniemi (2006), Hedetniemi (2006), and Ahmane, Bouchemakh, Sopena (2018).