Authors: Uéverton dos Santos Souza ^1,2,3; Fábio Protti ²; Maise Silva ⁴

• 1 Centro Federal de Educação Tecnológica Celso Suckow da Fonseca (Rio de Janeiro)
• 2 Instituto de Computação [Niteroi-Rio de Janeiro]
• 3 Centro Federal de Educação Tecnológica Celso Suckow da Fonseca [Rio de Janeiro] [CEFET/RJ]
• 4 Instituto de Ciência e Tecnologia [Rio das Ostras]

Flood-it is a combinatorial game played on a colored graph G whose aim is to make the graph monochromatic using the minimum number of flooding moves, relatively to a fixed pivot. Free-Flood-it is a variant where the pivot can be freely chosen for each move of the game. The standard versions of Flood-it and Free-Flood-it are played on m ×n grids. In this paper we analyze the behavior of these games when played on other classes of graphs, such as d-boards, powers of cycles and circular grids. We describe polynomial time algorithms to play Flood-it on C2n (the second power of a cycle on n vertices), 2 ×n circular grids, and some types of d-boards (grids with a monochromatic column). We also show that Free-Flood-it is NP-hard on C2n and 2 ×n circular grids.