Fanciful Figurines flip Free Flood-It -- Polynomial-Time Miniature Painting on Co-gem-free Graphs

Abstract

Inspired by the eponymous hobby, we introduce Miniature Painting as the computational problem to paint a given graph $G=(V,E)$ according to a prescribed template $t \colon V \rightarrow C$, which assigns colors $C$ to the vertices of $G$. In this setting, the goal is to realize the template using a shortest possible sequence of brush strokes, where each stroke overwrites a connected vertex subset with a color in $C$. We show that this problem is equivalent to a reversal of the well-studied Free Flood-It game, in which a colored graph is decolored into a single color using as few moves as possible. This equivalence allows known complexity results for Free Flood-It to be transferred directly to Miniature Painting, including NP-hardness under severe structural restrictions, such as when $G$ is a grid, a tree, or a split graph. Our main contribution is a polynomial-time algorithm for Miniature Painting on graphs that are free of induced co-gems, a graph class that strictly generalizes cographs. As a direct consequence, Free Flood-It is also polynomial-time solvable on co-gem-free graphs, independent of the initial coloring.