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On the complexity of the Maker-Breaker happy vertex game

Mathieu Hilaire, Perig Montfort, Nacim Oijid

TL;DR

The paper studies the Scoring Happy Vertex Game (SHVG), a Maker-Breaker variant where Maker maximizes red happy vertices, situating it within Milnor's universe and linking it to the domination game. It develops a framework using the literal-clause incidence graph and reductions from Quantified MAX-2-SAT to establish strong complexity results, including PSPACE-completeness on trees and NP-hardness on caterpillars, while providing polynomial-time results for subdivided stars and unions of paths and an FPT algorithm by neighborhood diversity. The main contributions are the hardness proofs via new 2-SAT variants, the precise scores on specific graph classes, and the algorithmic toolkit (super-lemma, h-function) that guides optimal play in this scoring setting. These results advance the understanding of scoring positional games on graphs and open avenues for further exploration of tractable graph families and parameterized approaches.

Abstract

Given a c-colored graph G, a vertex of G is happy if it has the same color as all its neighbors. The notion of happy vertices was introduced by Zhang and Li to compute the homophily of a graph. Eto, et al. introduced the Maker-Maker version of the Happy vertex game, where two players compete to claim more happy vertices than their opponent. We introduce here the Maker-Breaker happy vertex game: two players, Maker and Breaker, alternately color the vertices of a graph with their respective colors. Maker aims to maximize the number of happy vertices at the end, while Breaker aims to prevent her. This game is also a scoring version of the Maker-Breaker Domination game introduced by Duchene, et al. as a happy vertex corresponds exactly to a vertex that is not dominated in the domination game. Therefore, this game is a very natural game on graphs and can be studied within the scope of scoring positional games. We initiate here the complexity study of this game, by proving that computing its score is PSPACE-complete on trees, NP-hard on caterpillars, and polynomial on subdivided stars. Finally, we provide the exact value of the score on graphs of maximum degree 2, and we provide an FPT-algorithm to compute the score on graphs of bounded neighborhood diversity. An important contribution of the paper is that, to achieve our hardness results, we introduce a new type of incidence graph called the literal-clause incidence graph for 2-SAT formulas. We prove that QMAX 2-SAT remains PSPACE-complete even if this graph is acyclic, and that MAX 2-SAT remains NP-complete, even if this graph is acyclic and has maximum degree 2, i.e. is a union of paths. We demonstrate the importance of this contribution by proving that Incidence, the scoring positional game played on a graph is also PSPACE-complete when restricted to forests.

On the complexity of the Maker-Breaker happy vertex game

TL;DR

The paper studies the Scoring Happy Vertex Game (SHVG), a Maker-Breaker variant where Maker maximizes red happy vertices, situating it within Milnor's universe and linking it to the domination game. It develops a framework using the literal-clause incidence graph and reductions from Quantified MAX-2-SAT to establish strong complexity results, including PSPACE-completeness on trees and NP-hardness on caterpillars, while providing polynomial-time results for subdivided stars and unions of paths and an FPT algorithm by neighborhood diversity. The main contributions are the hardness proofs via new 2-SAT variants, the precise scores on specific graph classes, and the algorithmic toolkit (super-lemma, h-function) that guides optimal play in this scoring setting. These results advance the understanding of scoring positional games on graphs and open avenues for further exploration of tractable graph families and parameterized approaches.

Abstract

Given a c-colored graph G, a vertex of G is happy if it has the same color as all its neighbors. The notion of happy vertices was introduced by Zhang and Li to compute the homophily of a graph. Eto, et al. introduced the Maker-Maker version of the Happy vertex game, where two players compete to claim more happy vertices than their opponent. We introduce here the Maker-Breaker happy vertex game: two players, Maker and Breaker, alternately color the vertices of a graph with their respective colors. Maker aims to maximize the number of happy vertices at the end, while Breaker aims to prevent her. This game is also a scoring version of the Maker-Breaker Domination game introduced by Duchene, et al. as a happy vertex corresponds exactly to a vertex that is not dominated in the domination game. Therefore, this game is a very natural game on graphs and can be studied within the scope of scoring positional games. We initiate here the complexity study of this game, by proving that computing its score is PSPACE-complete on trees, NP-hard on caterpillars, and polynomial on subdivided stars. Finally, we provide the exact value of the score on graphs of maximum degree 2, and we provide an FPT-algorithm to compute the score on graphs of bounded neighborhood diversity. An important contribution of the paper is that, to achieve our hardness results, we introduce a new type of incidence graph called the literal-clause incidence graph for 2-SAT formulas. We prove that QMAX 2-SAT remains PSPACE-complete even if this graph is acyclic, and that MAX 2-SAT remains NP-complete, even if this graph is acyclic and has maximum degree 2, i.e. is a union of paths. We demonstrate the importance of this contribution by proving that Incidence, the scoring positional game played on a graph is also PSPACE-complete when restricted to forests.
Paper Structure (19 sections, 21 theorems, 11 equations, 3 figures)

This paper contains 19 sections, 21 theorems, 11 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Framework of the study
  3. Outline of the paper
  4. Preliminaries
  5. Definitions and Notations
  6. Scoring game
  7. Results inherited from the Maker-Breaker domination game
  8. Simplifying instances
  9. Hardness results
  10. Quantified MAX SAT restrictions
  11. Trees
  12. Caterpillars
  13. Efficient algorithms
  14. Union of paths
  15. Single path
  16. ...and 4 more sections

Key Result

Lemma 3

For any game state $(G, M, B)$, we have:

Figures (3)

  • Figure 1: The reduction in Definition \ref{['decomposed graph']}
  • Figure 2: Computing the score on $T_3$, the complete binary tree of depth $3$.
  • Figure 3: The literal-clause incidence graph $G_\varphi$ with $\varphi=(x_1\lor\lnot x_2)\land(x_2\lor x_3)\land(\lnot x_1\lor x_3)$

Theorems & Definitions (27)

  • Definition 2
  • Lemma 3
  • Definition 4
  • Theorem 5: Milnor milnor1953sums
  • Theorem 6: Milnor 1953 milnor1953sums
  • Remark 7
  • Corollary 8
  • Remark 9
  • Definition 10
  • Lemma 11
  • ...and 17 more