Extremal properties of flood-filling games

TL;DR

This paper investigates the maximum number of flooding operations needed to make any coloured graph monochromatic in the Flood-It game, providing bounds and tight results for trees and denser graphs.

Contribution

It offers the first systematic bounds on worst-case moves in Flood-It, including tight bounds for trees and improvements for graphs with higher edge-density.

Findings

Established upper and lower bounds for arbitrary graphs.

Proved bounds are tight for trees.

Analyzed how edge-density affects the maximum number of moves.

Abstract

The problem of determining the number of "flooding operations" required to make a given coloured graph monochromatic in the one-player combinatorial game Flood-It has been studied extensively from an algorithmic point of view, but basic questions about the maximum number of moves that might be required in the worst case remain unanswered. We begin a systematic investigation of such questions, with the goal of determining, for a given graph, the maximum number of moves that may be required, taken over all possible colourings. We give several upper and lower bounds on this quantity for arbitrary graphs and show that all of the bounds are tight for trees; we also investigate how much the upper bounds can be improved if we restrict our attention to graphs with higher edge-density.