On Complexity of Flooding Games on Graphs with Interval Representations

TL;DR

This paper studies the computational complexity of flooding games on graphs with interval representations, showing polynomial solvability with a fixed number of colors and NP-completeness when colors are unbounded.

Contribution

It characterizes the complexity of flooding games on interval and split graphs based on the number of colors, providing new complexity results.

Findings

Polynomial-time solvable with fixed number of colors on interval graphs

NP-complete when the number of colors is unbounded on proper interval graphs

Similar complexity results established for split graphs

Abstract

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.