Burning Numbers of Barbells

Abstract

Motivated by a discrete-time process intended to measure the speed of the spread of contagion in a graph, the burning number b(G) of a graph G, is defined as the smallest integer k for which there are vertices x₁,…,x_k such that for every vertex u of G, there exists i ∈ {1,…,k} with d_G(u, x_i) ≤ k − i, and d_G(x_i, x_j) ≥ j − i for any 1 ≤ i < j ≤ k. The graph burning problem has been shown to be NP-complete even for some acyclic graphs with maximum degree three. In this paper, we determine the burning numbers of all short barbells and long barbells, respectively.