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Complexity of Maker-Breaker Games on Edge Sets of Graphs

Eric Duchêne, Valentin Gledel, Fionn Mc Inerney, Nicolas Nisse, Nacim Oijid, Aline Parreau, Miloš Stojaković

TL;DR

It is proved that deciding who wins the perfect matching game and the H-game is PSPACE-complete, even for the latter in small-diameter graphs if $H$ is a tree.

Abstract

We study the algorithmic complexity of Maker-Breaker games played on the edge sets of general graphs. We mainly consider the perfect matching game and the $H$-game. Maker wins if she claims the edges of a perfect matching in the first, and a copy of a fixed graph $H$ in the second. We prove that deciding who wins the perfect matching game and the $H$-game is PSPACE-complete, even for the latter in small-diameter graphs if $H$ is a tree. Toward finding the smallest graph $H$ for which the $H$-game is PSPACE-complete, we also prove that such an $H$ of order 51 and size 57 exists. We then give several positive results for the $H$-game. As the $H$-game is already PSPACE-complete when $H$ is a tree, we mainly consider the case where $H$ belongs to a subclass of trees. In particular, we design two linear-time algorithms, both based on structural characterizations, to decide the winners of the $P_4$-game in general graphs and the $K_{1,\ell}$-game in trees. Then, we prove that the $K_{1,\ell}$-game in any graph, and the $H$-game in trees are both FPT parameterized by the length of the game, notably adding to the short list of games with this property, which is of independent interest. Another natural direction to take is to consider the $H$-game when $H$ is a cycle. While we were unable to resolve this case, we prove that the related arboricity-$k$ game is polynomial-time solvable. In particular, when $k=2$, Maker wins this game if she claims the edges of any cycle.

Complexity of Maker-Breaker Games on Edge Sets of Graphs

TL;DR

It is proved that deciding who wins the perfect matching game and the H-game is PSPACE-complete, even for the latter in small-diameter graphs if is a tree.

Abstract

We study the algorithmic complexity of Maker-Breaker games played on the edge sets of general graphs. We mainly consider the perfect matching game and the -game. Maker wins if she claims the edges of a perfect matching in the first, and a copy of a fixed graph in the second. We prove that deciding who wins the perfect matching game and the -game is PSPACE-complete, even for the latter in small-diameter graphs if is a tree. Toward finding the smallest graph for which the -game is PSPACE-complete, we also prove that such an of order 51 and size 57 exists. We then give several positive results for the -game. As the -game is already PSPACE-complete when is a tree, we mainly consider the case where belongs to a subclass of trees. In particular, we design two linear-time algorithms, both based on structural characterizations, to decide the winners of the -game in general graphs and the -game in trees. Then, we prove that the -game in any graph, and the -game in trees are both FPT parameterized by the length of the game, notably adding to the short list of games with this property, which is of independent interest. Another natural direction to take is to consider the -game when is a cycle. While we were unable to resolve this case, we prove that the related arboricity- game is polynomial-time solvable. In particular, when , Maker wins this game if she claims the edges of any cycle.
Paper Structure (17 sections, 20 theorems, 5 equations, 15 figures)

This paper contains 17 sections, 20 theorems, 5 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Complexity of positional games
  3. Complexity of short games
  4. Our contributions
  5. Preliminaries
  6. Hardness Results
  7. Perfect-matching game
  8. $H$-game
  9. Linear-time Algorithms
  10. Linear-time algorithm for the $P_4$-game
  11. Case 1. $G$ contains a diamond.
  12. Case 2. $G$ contains a triangle, but not a diamond.
  13. Case 3. $G$ contains an odd path between two vertices of degree at least 3, but not a triangle.
  14. Case 4. $G$ contains an odd cycle with a unique vertex of degree at least 3.
  15. $K_{1,\ell}$-game in trees
  16. ...and 2 more sections

Key Result

Lemma 2.1

Let $G'$ be a subgraph of $G$, ${\cal F} \subseteq 2^{E(G)}$ a set of winning sets, and ${\cal F}' \subseteq {\cal F} \cap 2^{E(G')}$. If Maker wins in $(E(G'),{\cal F}')$, then she wins in $(E(G),{\cal F})$.

Figures (15)

  • Figure 1: How parallel edges are removed in the perfect matching game in Lemma \ref{['remove double edge']}.
  • Figure 2: The variable gadget in the graph $G$ constructed in the proof of Theorem \ref{['thm:perfect-matching']}.
  • Figure 3: Construction for a variable $x_i$ in clauses $C^{i_1}, \dots, C^{i_{k_i}}$ in $\phi$ in the graph $G$ constructed in the proof of Theorem \ref{['thm:perfect-matching']}.
  • Figure 4: Construction for a clause $C_j = (x_{i_1} \vee \dots \vee x_{i_6})$ in $\phi$ in the graph $G$ constructed in the proof of Theorem \ref{['thm:perfect-matching']}.
  • Figure 5: The $y^i_{i_j}$ vertices in the graph $G$ constructed in the proof of Theorem \ref{['thm:perfect-matching']}.
  • ...and 10 more figures

Theorems & Definitions (43)

  • Lemma 2.1: Folklore
  • Lemma 2.2: Folklore
  • Theorem 2.3: Erdős-Selfridge Criterion Erdos-selfridge
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • ...and 33 more