The Flood Polynomial of a Graph

The flood polynomial of a simple finite graph is a weight generating function that counts all flooding cascade sets of the graph. The flood polynomial is inspired by the water mechanics in the video game Minecraft. We give necessary conditions for two graphs to have the same flood polynomial. We then provide a formula for the flood polynomials of certain families of graphs. We will see that many flood polynomials can be expressed using a Fibonacci-like recurrence and in some cases are equal to Fibonacci or Lucas polynomials. We then provide general examples of pairs of distinct graphs with the same flood polynomial. In these examples, the flood polynomial will be expressed as the product of Fibonacci and Lucas polynomials.