Fanciful Figurines flip Free Flood-It -- Polynomial-Time Miniature Painting on Co-gem-free Graphs
Christian Rosenke, Mark Scheibner

TL;DR
This paper introduces Miniature Painting, a graph coloring problem related to Free Flood-It, proving its NP-hardness in general but providing a polynomial-time solution for co-gem-free graphs, a broad class including cographs.
Contribution
The paper establishes the equivalence between Miniature Painting and Free Flood-It, proves NP-hardness on various graph classes, and presents a polynomial-time algorithm for co-gem-free graphs.
Findings
Miniature Painting is NP-hard on grids, trees, and split graphs.
Polynomial-time algorithm for Miniature Painting on co-gem-free graphs.
Free Flood-It is polynomial-time solvable on co-gem-free graphs.
Abstract
Inspired by the eponymous hobby, we introduce Miniature Painting as the computational problem to paint a given graph according to a prescribed template , which assigns colors to the vertices of . 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 . 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 is a grid, a tree, or a split graph. Our main contribution is a polynomial-time algorithm for…
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Taxonomy
TopicsAdvanced Graph Theory Research · Computational Geometry and Mesh Generation · Complexity and Algorithms in Graphs
