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Paintbucket on graphs is PSPACE-complete

Ethan J. Saunders, Peter Selinger

TL;DR

This work proves that Paintbucket on graphs, a natural generalization where players flip an opponent's connected color class on a simple graph, is PSPACE-complete. The authors reduce from avoider-enforcer games by constructing a polynomial-size bipartite graph \(G_{K}{(C, {\cal A})}\) with a parameter \(K \ge |C|+2\) that faithfully simulates any avoider-enforcer position \((C, {\cal A})\) via a parity-dependent argument. They introduce an intended play framework and show that any deviation (‘shenanigan’) leads to an immediate loss, ensuring the reduction's correctness through a robust simulation. They prove a key proposition linking the winner in \(G_{K}{(C, {\cal A})}\) to the winner in \((C, {\cal A})\) depending on the parity of \(|C|\), establishing PSPACE-hardness, whilePaintbucket on graphs remains in PSPACE. The paper concludes with an open question regarding the PSPACE status of the original grid-based Paintbucket version.

Abstract

The game of Paintbucket was recently introduced by Amundsen and Erickson. It is played on a rectangular grid of black and white pixels. The players alternately fill in one of their opponent's connected components with their own color, until the entire board is just a single color. The player who makes the last move wins. It is not currently known whether there is a simple winning strategy for Paintbucket. In this paper, we consider a natural generalization of Paintbucket that is played on an arbitrary simple graph, and we show that the problem of determining the winner in a given position of this generalized game is PSPACE-complete.

Paintbucket on graphs is PSPACE-complete

TL;DR

This work proves that Paintbucket on graphs, a natural generalization where players flip an opponent's connected color class on a simple graph, is PSPACE-complete. The authors reduce from avoider-enforcer games by constructing a polynomial-size bipartite graph \(G_{K}{(C, {\cal A})}\) with a parameter that faithfully simulates any avoider-enforcer position \((C, {\cal A})\) via a parity-dependent argument. They introduce an intended play framework and show that any deviation (‘shenanigan’) leads to an immediate loss, ensuring the reduction's correctness through a robust simulation. They prove a key proposition linking the winner in \(G_{K}{(C, {\cal A})}\) to the winner in \((C, {\cal A})\) depending on the parity of , establishing PSPACE-hardness, whilePaintbucket on graphs remains in PSPACE. The paper concludes with an open question regarding the PSPACE status of the original grid-based Paintbucket version.

Abstract

The game of Paintbucket was recently introduced by Amundsen and Erickson. It is played on a rectangular grid of black and white pixels. The players alternately fill in one of their opponent's connected components with their own color, until the entire board is just a single color. The player who makes the last move wins. It is not currently known whether there is a simple winning strategy for Paintbucket. In this paper, we consider a natural generalization of Paintbucket that is played on an arbitrary simple graph, and we show that the problem of determining the winner in a given position of this generalized game is PSPACE-complete.

Paper Structure

This paper contains 10 sections, 8 theorems, 5 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Related work.
  3. Avoider-enforcer games
  4. Paintbucket on bipartite graphs
  5. Paintbucket is PSPACE-complete
  6. Statement of the main result
  7. Translation from avoider-enforcer games to Paintbucket
  8. Intended play, shenanigans, and simulation
  9. Proof of the main result
  10. Conclusion

Key Result

Lemma 3.1

Consider Paintbucket played on a bipartite graph $G$ with a distinguished white vertex $v$ to which $k$ black leaves are attached, as in Figure fig:claw. Let $m$ be the number of white vertices of $G$. Then if $m\leqslant k$, Black has a first-player winning strategy. Moreover, if $m<k$, Black has a

Figures (4)

  • Figure 1: (a) An example game of Paintbucket played by the authors. White makes the last move and therefore wins. (b) The same game, played on a graph. (c) The same game, played on a bipartite graph.
  • Figure 2: A bipartite graph with $k$ black leaves attached to a white vertex.
  • Figure 3: The graph $G_{K}{(C,{\cal A})}$. Here, we have assumed that $A_1=\{c_1,c_2\}$, $A_2=\{c_2,c_3\}$, and $A_J=\{c_3,c_I\}$.
  • Figure 4: Schematic representation of the proof of Proposition \ref{['pro:main']}

Theorems & Definitions (19)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 4.1
  • ...and 9 more