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

TL;DR

The paper studies Miniature Painting, where a graph $G=(V,E)$ must be painted to a template $t:V\to C$ using the fewest brush strokes, each stroke painting a connected area $A$ with a color $c$. It establishes a precise equivalence to Free Flood-It: $G$ can be painted in $s$ strokes iff flooding from $t$ requires $s-1$ moves, enabling transfer of NP-hardness results to Miniature Painting across many graph classes. The main advance is a polynomial-time algorithm for Miniature Painting on co-gem-free graphs by exploiting a two-phase, canonical painting plan centered on a small dominating induced $P_4$ hub $D$ with $|D|=4$ and a bounded tail length $k\le 12$, leading to a polynomial-time solution for Free Flood-It on this class as well. Additionally, the approach yields a clean polynomial-time solution for cographs and provides a general structural framework (canonical painting plans) that could be extended to broader graph classes and tractable cases of Free Flood-It.

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.

Figures (1)

• Figure 1: (a) a graph model with template, (b) the path $P_4$ on four vertices, (c) the co-gem

Theorems & Definitions (32)

• Lemma 0
• Theorem 1
• Corollary 2: see Clifford2012FELLOWS2015Fukui2013
• Theorem 3
• Lemma 3
• Lemma 3
• Lemma 3
• Lemma 3
• Theorem 4
• proof