Burning number of theta graphs

Abstract

The burning number b(G) of a graph G was introduced by Bonato, Janssen, and Roshanbin [Lecture Notes in Computer Science 8882(2014)] to measure the speed of the spread of contagion in a graph. The graph burning problem is NP-complete even for trees. In this paper, we show that the burning number of any theta graph of order $n=q^2+r$ with $1\le r\le 2q+1$ is either q or $q+1$. Furthermore, we characterize all theta graphs that have burning number q or $q+1$.

Introduction

The burning number of a graph was first introduced by Bonato et al. [6] in 2014. It is related to contract processes on graphs such as the firefighter problem [3], [9], [11], graph cleaning problem [1] and graph bootstrap percolation [2]. Given a graph G, the burning process on G is a discrete-time process defined as follows. Initially, at time $t=0$ all vertices are unburned. At each time step t ≥ 1, one new unburned vertex is chosen to burn (if such a vertex is available); such a vertex is called a source of fire. If a vertex is burned, then it remains in that state until the end of the process. Once a vertex is burned in step t, in step $t+1$ each of its unburned neighbors becomes burned. The process ends when all vertices of G are burned (that is, let T be the smallest positive integer such that there is at least one vertex not burning in step $T-1$ and all vertices are burned in step T).

The burning number of a graph G, denoted by b(G), is the minimum number of steps needed for the process to end. Note that with our notation, $b\left(G\right)=T$.

Suppose that in the process of burning a graph G, we eventually burned the whole graph in k steps, and for each i (1 ≤ i ≤ k), we denote the vertex that we burn in the ith step by x_i. The sequence $(x_1,x_2,\dots ,x_k)$ is called a burning sequence for G. The burning number of G is the length of the shortest burning sequence for G, such a burning sequence is referred to as optimal. For example, for the path P₄ (or resp., P₅) with vertices {v₁, v₂, v₃, v₄} (or resp., {v₁, v₂, v₃, v₄, v₅}), the sequence (v₂, v₄) (or resp., (v₂, v₄, v₅)) is an optimal burning sequence; see Fig. 1.

In 2016, Bonato et al. [6] studied the burning number of paths and cycles, and based on these results, they made a conjecture on the upper bound for the burning number.

Theorem 1.1 [6]

For a path P_n or a cycle C_n of order n, we have that $b\left(P_n\right)=b\left(C_n\right)=\lceil \sqrt{n}\rceil$. Moreover, if G is a graph of order n with a Hamiltonian (that is, a spanning) path, then $b\left(G\right)\le \lceil \sqrt{n}\rceil$.

Conjecture 1.2 [6]

For a connected graph G of order n, $b\left(G\right)\le \lceil \sqrt{n}\rceil$.

Bonato et al. [6] showed that $b\left(G\right)\le 2\lceil \sqrt{n}\rceil -1$. Bessy et al. [5] improved the upper bound to $b\left(G\right)\le (\sqrt{\frac{32}{19}}+o\left(1\right))\sqrt{n}$. Recently, Land and Lu [13] proved that $b\left(G\right)\le \lceil \frac{-3+\sqrt{24n+33}}{4}\rceil$.

It is shown that the burning problem is NP-complete even for trees and path-forests (that is, disjoint unions of paths), see [4], [15]. Then it is interesting to determine the burning number of special classes of graphs. Bonato and Lidbetter [8] considered the bounds on the burning numbers of spiders (which are trees with exactly one vertex of degree strictly greater than two) and path-forests. Sim et al. [16] studied the burning number of generalized Petersen graphs. Fitzpatrick and Wilm [12] determined the burning number of circulant graphs. Recently, Mitsche et al. [14] focused on a few probabilistic aspects of the burning problem. For more on graph burning and graph searching, see the recent book [10].

In this paper, we show that the burning number of the theta graph of order $n=q^2+r$ with $1\le r\le 2q+1$ is either q or $q+1$. Furthermore, we find all the sufficient conditions for a theta graph to have burning number q or $q+1$.