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, $P_4$-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 $P_4$ -- 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 $P_5$, 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, $P_5$, and $C_5$, 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.

A short version of this paper will appear in the proceedings of the 51st International Symposium on Mathematical Foundations of Computer Science (MFCS 2026)