The Flood Polynomial of a Graph
Karin R. Saoub, Michael Weselcouch, Trey Wilhoit, Jackson Wills
TL;DR
The paper introduces the flood polynomial $F_G(x)$ as a generating function counting flooding cascade sets in a finite graph, inspired by Minecraft water dynamics. It develops the basic theory, including multiplicativity under disjoint unions and recoverability of vertex counts from $F_G(x)$, and derives recursive formulas for flood polynomials on several graph families such as paths, cycles, and triangle mosaics, highlighting connections to Fibonacci and Lucas polynomials. It also shows that distinct graphs can share the same flood polynomial and provides structural decompositions that express many flood polynomials as products of Fibonacci and Lucas polynomials. The work lays groundwork for a broader study of flood polynomials, poses open questions about extensions to new graph families, and explores how this invariant encodes combinatorial flooding behavior with potential applications in graph theory and combinatorics.
Abstract
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.