Flood-It is a single-player game aimed at making a precolored graph $G$ 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 can be chosen anew in every move. Determining if a graph can be made monochromatic in at most $k$ moves is NP-complete for both variants. This complexity remains even under strong structural restrictions, such as split graphs and trees. The Free Flood-It variant is generally deemed harder, as it remains challenging across several graph classes where the fixed-pivot variant becomes tractable, including co-comparability and AT-free graphs. Cographs, or $P_4$-free graphs, are among the few classes where even Free Flood-It is solvable in polynomial time, making them our starting point.
We consider the ten natural one-vertex extensions of $P_4$, termed jewels, and investigate the complexity of both flooding games across the $1024$ graph classes obtained by forbidding certain subsets of these graphs as induced subgraphs. Our main contribution is a polynomial-time algorithm for Free Flood-It on graphs free of the three jewels bull, gem, and $P_5$, covering $128$ of the $1024$ classes. Additionally, we prove that both variants remain NP-complete on thin-spider graphs, which exclude the eight jewels banner, co-banner, chair, gem, house, kite, $P_5$, and $C_5$, establishing hardness for an additional $256$ classes. Together with known algorithms and hardness results, our work clarifies the complexity of both Flood-It variants for $896$ of the $1024$ considered graph classes.