Burning numbers of t-unicyclic graphs

TL;DR

This paper investigates the burning number, a measure of how quickly a graph can be 'burned' through a sequence of vertices, focusing on t-unicyclic graphs and establishing bounds and exact values for certain cases.

Contribution

It provides bounds for the burning number of t-unicyclic graphs and determines exact values for cases where t is less than or equal to 2.

Findings

Established bounds for the burning number of t-unicyclic graphs.

Determined the burning number for all t-unicyclic graphs with t ≤ 2.

Connected the burning numbers of linear forests to those of t-unicyclic graphs.

Abstract

Given a graph $G$, the burning number of $G$ is the smallest integer $k$ for which there are vertices $x_1, x_2,\ldots,x_k$ such that $(x_1,x_2,\ldots,x_k)$ is a burning sequence of $G$. It has been shown that the graph burning problem is NP-complete, even for trees with maximum degree three, or linear forests. A $t$-unicyclic graph is a unicycle graph with exactly one vertex of degree greater than $2$. In this paper, we first present the bounds for the burning number of $t$-unicyclic graphs, and then use the burning numbers of linear forests with at most three components to determine the burning number of all $t$-unicyclic graphs for $t\le 2$.