  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 these
new coloring games on more general boards, or general graphs. 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. When the number of colors is a
fixed constant, these games can be solved in polynomial time on an interval
graph. On the other hand, if the number of colors is not bounded, the flooding
game is NP-complete on a proper interval graph. We also state similar results
for split graphs.
