Flood-it on AT-Free Graphs

TL;DR

This paper proves that computing the minimal number of moves in the Flood-it game can be done efficiently on AT-free graphs, expanding understanding of the game's complexity on specific graph classes.

Contribution

It establishes polynomial-time computability of Flood-it's minimal moves on AT-free graphs, a significant complexity result for this combinatorial game.

Findings

Polynomial-time algorithm for Flood-it on AT-free graphs

Complexity classification of Flood-it on specific graph classes

Advances understanding of game complexity in graph theory

Abstract

Solitaire {\sc Flood-it}, or {\sc Honey-Bee}, is a game played on a colored graph. The player resides in a source vertex. Originally his territory is the maximal connected, monochromatic subgraph that contains the source. A move consists of calling a color. This conquers all the nodes of the graph that can be reached by a monochromatic path of that color from the current territory of the player. It is the aim of the player to add all vertices to his territory in a minimal number of moves. We show that the minimal number of moves can be computed in polynomial time when the game is played on AT-free graphs.