Maker-Breaker Happy Vertex Game

• The Maker–Breaker Happy Vertex Game is a scoring positional game where Maker aims to maximize 'happy' Red vertices by coloring graph vertices, while Breaker seeks to thwart this strategy.
• It extends traditional domination games by linking vertex color agreement (homophily) with graph domination properties and enabling reductions from logical formulas.
• Key computational results include PSPACE-completeness on trees, NP-hardness on caterpillars, and fixed-parameter tractability for graphs with bounded neighborhood diversity.

The Maker–Breaker Happy Vertex Game (SHVG) is a scoring positional game on graphs, central to the study of coloring-based homophily measures and the algorithmic complexity of adversarial graph processes. In the SHVG, two players—Maker (Red) and Breaker (Blue)—alternately color uncolored vertices of a simple undirected graph $G=(V,E)$ with their respective colors. A vertex is deemed “happy” if its color matches all its neighbors. Maker’s aim is to maximize the number of happy Red vertices at the end of the game, while Breaker seeks to minimize this total. The SHVG extends foundational concepts from homophily in networks and the Maker–Breaker domination game framework, and poses deep questions about algorithmic tractability, parameterized complexity, and the construction of graph gadgets for hardness reductions (Hilaire et al., 12 Jan 2026).

1. Formal Structure and Notation

Given a simple undirected graph $G=(V,E)$, two players alternately select and color uncolored vertices, with Maker (Red) and Breaker (Blue) using exclusive colors. Once all vertices are colored, a vertex $v$ is happy if every neighbor of $v$ is of the same color as $v$. Maker’s objective is to maximize the final count of happy Red vertices; Breaker aims to minimize this.

The standard scoring-game formalism is adopted: a game position is given by the tuple $(G, M, B)$, where $M\subseteq V$ are Red-colored vertices, $B\subseteq V$ are Blue-colored vertices, and the set of free vertices is $V_F = V \setminus (M \cup B)$. The recursive score functions are defined as:

• $Ms(G, M, B) = \max_{x\in V_F} Bs(G, M\cup\{x\}, B)$ (Maker’s turn),
• $Bs(G, M, B) = \min_{x\in V_F} Ms(G, M, B\cup\{x\})$ (Breaker’s turn),

with the terminal case $s(G,M,B) = |\{ v\in V : N[v] \subseteq M \}|$ (the count of Red-happy vertices). The score of the game is $s(G) = Ms(G,\emptyset,\emptyset)$ if Maker starts, or $s(G)=Bs(G,\emptyset,\emptyset)$ if Breaker starts.

2. Connections to Homophily and Domination Games

The notion of “happy vertex” originates from homophily-based graph analysis, where it provides a measure of local agreement of colors introduced by Zhang & Li (2015). This game-theoretic operationalization is significant for quantifying network cohesion under adversarial conditions.

SHVG further serves as a scoring extension of the Maker–Breaker domination game (Duchêne et al., 2020), where players alternately claim vertices, and Maker attempts to assemble a dominating set. In SHVG, the number of Maker’s happy vertices precisely equals the number of vertices not dominated by Breaker. As such, the outcomes of this game encode domination properties:

• $s(G) = 0$ if and only if Breaker has a domination-winning strategy ($\mathcal{D}$ outcome).
• $s(G) > 0$ if and only if Maker can force at least one happy vertex ($\mathcal{N}$ or $\mathcal{S}$ outcome).

This correspondence enables reduction between SHVG and classical domination game complexities, providing a bridge for transferring hardness.

3. Computational Complexity and Algorithmic Tractability

The computational study of SHVG centers on the decision problem: given $G$ and $s$, decide if $s(G)\geq s$.

Hardness Results:

• SHVG is PSPACE-complete on trees (Theorem 3.1).
• SHVG is NP-hard on caterpillars (trees whose non-leaf subgraph is a path) (Theorem 3.2).

Tractable Cases:

• On subdivided stars (a central vertex with $k$ attached paths), Maker's score is $Ms = \lfloor l/2 \rfloor$ where $l$ is the number of arms with odd length (unless $l=0$, when $Ms=1$) and Breaker's best outcome is always $0$. The computation is $O(k)$ (Theorem 3.3a).
• For graphs of maximum degree 2 (disjoint union of paths and cycles), writing $l$ for the number of odd-length path components, Maker’s and Breaker’s optimal scores are $Ms(G) = \lceil l/2 \rceil$ and $Bs(G) = \lfloor l/2 \rfloor$, computable in linear time $O(|V|+|E|)$ (Theorem 3.3b).
• For graphs with neighborhood diversity $w$, the score can be computed in $O(3^w\cdot(|V|+|E|))$ time. Thus, SHVG is fixed-parameter tractable (FPT) parameterized by neighborhood diversity (Theorem 3.4).

Summary Table: Complexity Frontier

Graph Class	Complexity	Structural Restriction
Trees	PSPACE-complete	Arbitrary tree
Caterpillars	NP-hard	Tree with path backbone
Subdivided stars	Polynomial	Central vertex + arms
Degree-2 graphs (paths/cycles)	Polynomial	Max degree 2
Bounded neighborhood diversity	FPT	Partition into $w$ modules

This breakdown underscores a sharp boundary between hard and tractable graph classes, informed by topological features and modular structure.

4. The Literal–Clause Incidence Graph and Hardness Reductions

The reduction from Q-MAX-2-SAT and MAX-2-SAT is predicated on the “literal–clause incidence graph” for 2-CNF formulas. This construction, which is distinct from the classical bipartite variable-clause graph, defines vertices $x_i^+,x_i^-$ for each variable, and introduces an edge for every clause $(\ell_p \vee \ell_q)$ between the corresponding literals.

The key results are:

• QMAX 2-SAT remains PSPACE-complete even when the incidence graph is acyclic (Lemma 4.1).
• MAX 2-SAT remains NP-complete even when the incidence graph is acyclic with maximum degree 2 (Lemma 4.2)—thus, the incidence graph is a union of paths.

These constructions enable reductions to trees and caterpillars by encoding QBF assignment order via chains of literal gadgets, clause gadgets (short paths), and leaf-multiplicity to enforce play order. The gadgetization ensures that Maker achieves the target score if and only if the original formula is true, establishing the required complexity results (Hilaire et al., 12 Jan 2026).

5. Algorithmic Methods: Special Graphs and Parameterization

Degree-2 Graphs (Disjoint Union of Paths and Cycles):

1. Decompose $G$ into components. Cycles contribute zero to the score.
2. Count $l =$ number of odd-length paths.
3. Maker’s optimal score is $\lceil l/2 \rceil$, Breaker’s is $\lfloor l/2 \rfloor$. Time: $O(|V|+|E|)$.

Subdivided Stars:

• Input: Center $C$ with $k$ arms $P_i$ of lengths $n_i$.
• Compute $l =$ count of odd-length arms.
• If $l>0$, $Ms = \lfloor l/2 \rfloor$, $Bs=0$; if $l=0$, $Ms=1$, $Bs=0$.

FPT by Neighborhood Diversity:

1. Partition $V$ into $w$ modules (identical open neighborhoods).
2. Collapse each module to at most one free representative by the Super-Lemma.
3. For $3^w$ possible "color/free" assignments, recursively compute $Ms/Bs$.
4. Each terminal assignment scored in $O(n+m)$, for total $O(3^w(n+m))$.

These results leverage combinatorial reductions (e.g., Pairing Lemma, Super-Lemma) to simplify instances structurally, especially when modules or repetitive substructures can be collapsed without affecting the core min–max recursion (Hilaire et al., 12 Jan 2026).

6. Illustrative Examples

Example: Path on 3 Vertices $P_3$

Maker (Red) starts:

• Maker colors $v_2$ (R).
• Breaker (Blue) colors $v_1$ (B).
• Maker colors $v_3$ (R).

Result: vertices B–R–R; only $v_3$ is happy (score 1).

Example: Disjoint Union $P_3\cup P_3$

Two odd-length paths ($l=2$): $Ms = \lceil 2/2 \rceil = 1$, $Bs = \lfloor 2/2 \rfloor = 1$.

• Maker plays in one component; Breaker can prevent a happy vertex there, forcing Maker’s only happy vertex to be in the other component.

Example: Hardness Gadget (QBF Encoding)

Given a 2-CNF QBF, its literal-clause incidence forest is duplicated and chained; clause edges are replaced by paths via auxiliary gadgets, and leaves are added to literal-vertices in sufficient multiplicity to enforce assignment order. The achieved score coincides with QBF truth, and the instance is a tree.

7. Significance and Research Directions

The SHVG establishes a rigorous framework for scoring positional games on graphs, making it a pivotal instance in the computational study of adversarial coloring and homophily quantification. The identification of the literal–clause incidence graph yields new avenues for translating logical complexity into graph-theoretic hardness, as well as new kinds of reductions beyond traditional bipartite or hypergraph incidence models.

The tractability frontiers mapped out for bounded-degree, star-like, and module-based graphs suggest a nuanced landscape where specific structural graph parameters—such as neighborhood diversity—allow fixed-parameter tractable solutions, while minor topological increases (from star to caterpillar) elicit NP-hardness or PSPACE-completeness.

A plausible implication is that other scoring and threshold games derived from coloring or domination themes may admit similarly sharp dichotomies. Further research could focus on parameterized complexity under different invariants (e.g., clique-width, treewidth) and on variants involving more colors, biased moves, or weighted scores (Hilaire et al., 12 Jan 2026).