The 2-burning number of a graph

TL;DR

This paper introduces the 2-burning number in graphs, analyzing how information spreads from multiple sources over rounds, and provides exact values and bounds for specific graph classes and Cartesian products.

Contribution

It defines and studies the 2-burning number and sources in graphs, extending the burning number concept to multiple sources and rounds, with new results for specific graph classes.

Findings

Determined $b_2(G)$ and $t_2(G)$ for spiders and wheels.

Showed differences in parameters between graph classes.

Provided upper bounds for Cartesian product graphs.

Abstract

We study a discrete-time model for the spread of information in a graph, motivated by the idea that people believe a story when they learn of it from two different origins. Similar to the burning number, in this problem, information spreads in rounds and a new source can appear in each round. For a graph $G$, we are interested in $b_2(G)$, the minimum number of rounds until the information has spread to all vertices of graph $G$. We are also interested in finding $t_2(G)$, the minimum number of sources necessary so that the information spreads to all vertices of $G$ in $b_2(G)$ rounds. In addition to general results, we find $b_2(G)$ and $t_2(G)$ for the classes of spiders and wheels and show that their behavior differs with respect to these two parameters. We also provide examples and prove upper bounds for these parameters for Cartesian products of graphs.