An algorithmic analysis of Flood-It and Free-Flood-It on graph powers

Uéverton dos Santos Souza (1, 2) , Fábio Protti (2) , Maise Dantas Da Silva (3)

1 CEFET/RJ - Centro Federal de Educação Tecnológica Celso Suckow da Fonseca [Rio de Janeiro]
2 IC-UFF - Instituto de Computação [Niteroi-Rio de Janeiro]
3 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.

Mots clés: Combinatorial Games; Graph algorithms; Computational Complexity; Flood-it; Free-Flood-it

Domaines: Mathématique discrète [cs.DM]

Uéverton dos Santos Souza, Fábio Protti, Maise Dantas Da Silva. An algorithmic analysis of Flood-It and Free-Flood-It on graph powers. Discrete Mathematics and Theoretical Computer Science, 2014, Vol. 16 no. 3 (3), pp.279--290. ⟨10.46298/dmtcs.2086⟩. ⟨hal-01188900⟩