On Complexity of Flooding Games on Graphs with Interval Representations

The flooding games, which are called Flood-It, Mad Virus, or HoneyBee, are a kind of coloring games and they have been becoming popular online. In these games, each player colors one specified cell in his/her turn, and all connected neighbor cells of the same color are also colored by the color. This flooding or coloring spreads on the same color cells. It is natural to consider the coloring games on general graphs. That is, once a vertex is colored, the flooding flows along edges in the graph. Recently, computational complexities of the variants of the flooding games on several graph classes have been studied. In this paper, we investigate the flooding games on some graph classes characterized by interval representations. Our results state that the number of colors is a key parameter to determine the computational complexity of the flooding games. If the number of colors is not bounded, the flooding game is NP-complete even on caterpillars and proper interval graphs. On the other hand, when the number of colors is a fixed constant, the game can be solved in polynomial time on interval graphs. We also state similar results for split graphs.