Flood-It with Jewelry - Characterizing the Game Complexity for Cograph Generalizations

Abstract

Flood-It is a single-player game played on a precolored graph G, where the objective is to make G monochromatic using as few flooding moves as possible. In each move, a color c is selected and all vertices reachable from a fixed pivot vertex via a monochromatic path are recolored with c. In the free variant, the pivot may be chosen anew in every move.
Deciding whether a graph can be made monochromatic in at most k moves is NP-complete for both variants, fixed and free. This hardness persists even under strong structural restrictions such as split graphs and trees. The Free Flood-It variant is generally considered more difficult than its fixed-pivot counterpart, as it remains hard on several graph classes where the latter becomes tractable, including co-comparability and AT-free graphs.
Cographs, that is, P₄-free graphs, are among the few classes on which even Free Flood-It is solvable in polynomial time and therefore serve as our starting point. We consider the ten natural one-vertex extensions of P₄ - referred to as jewels - and study the complexity of both flooding games on the 1024 graph classes obtained by forbidding subsets of these graphs as induced subgraphs.
Our main contribution is a polynomial-time algorithm for Free Flood-It on graphs that are free of the three jewels bull, gem, and P₅, covering 128 of the 1024 classes. In addition, we prove that both variants remain NP-complete on thin-spider graphs, which exclude the eight jewels banner, co-banner, chair, gem, house, kite, P₅, and C₅, thereby establishing hardness for 256 additional classes.
Combined with known algorithms and hardness results, our work determines the complexity of both Flood-It variants for 896 of the 1024 considered graph classes.