Department of Mathematical Sciences
Discrete Math Seminar
Tuesday, November 11th, 2025
4:00PM-4:50PM
Olin Hall 126
Speaker: Ralihe Villagran, WPI
Title: On the zero forcing and the lazy burning numbers of a graph.
Abstract: We will give an introduction to these two concepts. Given a graph G, they both involve a dynamical process carried out on the graph.
Let Z be a subset of the vertices of G. Assume the vertices in Z are colored blue and the vertices in the complement are colored white. Then the color change process described by the rule: "If a vertex z in Z has a unique white neighbor w, then change the color of w to blue." Is known as the zero-forcing process. Moreover, Z is known as a zero-forcing set if the color change rule can be applied until the whole vertex set is blue, and the zero-forcing number, denoted by Z(G) is the minimum size of a zero-forcing set.
On the other hand, in the burning process, each node is either burned or unburned; if a node is burned, then it remains in that state until the end of the process. In every round, choose one additional unburned node to burn (if such a node is available). Once a node is burned in round t, in round t+1, each of its unburned neighbors becomes burned. The process ends when all nodes are burned. The burning number of a graph G, written by b(G), is then defined as the minimum number of rounds needed for the process to end.
We will see how a relationship between these two concepts has been established, involving hypergraphs.