Maker-Breaker Domination Games

• Maker-Breaker Domination Games are defined as positional contests on graphs where players alternate selecting vertices, with Dominator aiming for a dominating set and Staller seeking to isolate a vertex.
• The framework extends classical domination by incorporating variants such as disjoint, total, and biased games, along with quantitative invariants that measure win-speed.
• Structural tools like pairing dominating sets, hypergraph dualities, and product constructions yield precise outcome characterizations and drive complexity and algorithmic analyses.

Maker-Breaker domination games are positional games on the vertex set of a graph in which Dominator seeks to occupy a dominating set while Staller seeks to prevent this, typically by isolating a vertex through occupying an entire closed neighborhood. Since their introduction as Maker-Breaker positional games by Duchêne, Gledel, Parreau, and Renault, the subject has developed into a family of graph games that includes the standard Maker-Breaker domination game, the disjoint domination game, total-domination analogues, biased games, predomination, and several criticality notions, together with quantitative invariants that measure how fast either player can force a win (Duchêne et al., 2018, Gledel et al., 2018, Bagan et al., 13 Jan 2026).

1. Formal game model and hypergraph formulations

Let $G=(V,E)$ be a finite simple graph. In the standard Maker-Breaker domination game, Dominator and Staller alternately select previously unplayed vertices of $G$. The game is played either as a D-game, where Dominator starts, or as an S-game, where Staller starts. Dominator wins as soon as the vertices he has selected form a dominating set of $G$, that is, a set $D\subseteq V$ with $N[D]=V$. Staller wins as soon as she has occupied all vertices of the closed neighborhood $N[v]$ of some vertex $v$, equivalently as soon as she isolates a vertex (Duchêne et al., 2018).

The standard outcome notation is $o(G)\in\{D,N,S\}$, where $D$ means that Dominator wins whether he starts or moves second, $S$ means that Staller wins whether she starts or moves second, and $G$0 means that the first player to move has a winning strategy. A standard imagination argument implies that the outcome $G$1 does not occur in Maker-Breaker games, so $G$2 (Duchêne et al., 2018).

Two hypergraph viewpoints recur throughout the literature. In one, Staller acts as Maker on the closed-neighborhood hypergraph

$G$3

trying to occupy an entire hyperedge, while Dominator acts as Breaker and tries to claim a transversal. In the other, Dominator acts as Maker on a hypergraph whose hyperedges are dominating sets, or in some formulations minimum dominating sets. This duality is central in both structural arguments and complexity reductions (Bujtás et al., 2022, Bagan et al., 13 Jan 2026).

2. Quantitative invariants and order relations

If Dominator has a winning strategy in the D-game, the Maker-Breaker domination number $G$4 is the minimum number of Dominator moves needed to force a win; if he cannot force a win, then $G$5. Analogously, $G$6 is defined for the S-game. The fast-winning parameters for Staller are $G$7 and $G$8, which denote the minimum numbers of Staller moves needed to force a win in the D-game and S-game, respectively, with the convention $G$9 if she has no winning strategy (Gledel et al., 2018, Bujtás et al., 2022).

These parameters satisfy several basic inequalities. Since every total dominating set is a dominating set and playing second cannot help Dominator in the ordinary domination setting, one has

$G$0

On Staller’s side,

$G$1

because Staller must eventually claim all $G$2 vertices of some closed neighborhood, and playing first never hurts her (Gledel et al., 2018, Bujtás et al., 2022).

A notable feature of the Maker-Breaker setting is that the gaps between these parameters can be large. For any integers $G$3, there exists a graph $G$4 with

$G$5

Similarly, for every three integers $G$6 with $G$7, there exists a graph $G$8 such that

$G$9

These realizability results distinguish Maker-Breaker domination from the usual game-domination parameters, where comparable gaps are much more constrained (Gledel et al., 2018, Bujtás et al., 2022).

3. Structural methods and graph classes

A central sufficient condition for $D\subseteq V$0 is the existence of a pairing dominating set. This is a collection of disjoint pairs

$D\subseteq V$1

such that

$D\subseteq V$2

If such a structure exists, Dominator can play a matching-reply strategy: whenever Staller plays one vertex of a pair, Dominator immediately plays its mate. This guarantees that Dominator’s claimed vertices form a dominating set (Duchêne et al., 2018).

This pairing viewpoint extends to several graph classes. On cographs, $D\subseteq V$3 if and only if $D\subseteq V$4 admits a pairing dominating set. On trees, repeated stripping of pendant $D\subseteq V$5’s yields a linear-time decision procedure; in particular, $D\subseteq V$6 on a tree exactly when the tree has a perfect matching. Residual graphs also yield exact formulas for $D\subseteq V$7 and $D\subseteq V$8 for arbitrary trees (Duchêne et al., 2018, Gledel et al., 2018).

A second major method is partition into basic subgraphs. If $D\subseteq V$9 admits a perfect $N[D]=V$0-factor, meaning a spanning subgraph whose components are edges or cycles, then $N[D]=V$1. Hall’s argument implies that every $N[D]=V$2-regular graph satisfies Hall’s condition on the associated incidence graph, hence every $N[D]=V$3-regular graph admits a perfect $N[D]=V$4-factor and so $N[D]=V$5. For outerplanar graphs and block graphs, $N[D]=V$6 if and only if $N[D]=V$7 admits a perfect $N[D]=V$8-factor, which yields polynomial-time algorithms in these classes (Bagan et al., 2024).

Interval graphs admit a sharper characterization: $N[D]=V$9 if and only if $N[v]$0 admits a pairing dominating set, and every pairing dominating set in an interval graph can be transformed into an adjacent pairing dominating set. This places the second-player decision problem for interval graphs in NP, and for $N[v]$1-nested interval graphs there is an $N[v]$2 algorithm for deciding the existence of such a structure (Bagan et al., 2024).

Product constructions furnish another source of structural transfer. If both $N[v]$3 and $N[v]$4 admit nontrivial path covers, then Dominator wins in both the D-game and the S-game on $N[v]$5. For corona products, if $N[v]$6, then

$N[v]$7

These product theorems reduce the game on large composite graphs to the game on small grids or on the second factor alone (Dokyeesun, 2023, Divakaran et al., 2024).

4. Major variants of the domination framework

Several distinct games now fall under the label “Maker-Breaker domination games,” differing in the target structure, bias, or initial conditions.

Variant	Winning target	Representative result
Standard MBD game	Dominator occupies a dominating set	Deciding the winner is PSPACE-complete even on bipartite and split graphs (Duchêne et al., 2018)
Disjoint domination game	Create or avoid two disjoint dominating sets	Maker wins on every connected graph in the Breaker-start nonbiased game, and in the $N[v]$8 game on every connected graph regardless of who starts (Bujtás et al., 2014)
MBTD game	Dominator occupies a total dominating set	There are infinitely many connected cubic graphs in which Staller wins (Gledel et al., 2019)
Biased MBD game	Dominator selects $N[v]$9 vertices and Staller $v$0 vertices per move	For any claw-free graph $v$1, $v$2 (Brešar et al., 14 Mar 2025)
Predominated MBD game	Dominate $v$3 on a predominated graph $v$4	MBD-critical trees are characterized, and recognition is linear-time (Bujtás et al., 14 Mar 2025)

The disjoint domination game, introduced by Bujtás and Tuza, is a Maker-Breaker type game in which the issue is to create or avoid two disjoint dominating sets in graphs without isolated vertices. In the $v$5 Breaker-start version, Maker has a winning strategy on all connected graphs. The proof uses the Opposite Neighbor Strategy, and the same argument implies that in the $v$6 game Maker wins on every connected graph regardless of who starts. The Bicolored Domination Game, a restricted nonbiased variant in which Maker may only use one color and Breaker the other, is Maker-win on every graph without isolated vertices (Bujtás et al., 2014).

The Maker-Breaker total domination game replaces domination by total domination. Dominator now seeks a total-dominating set, while Staller wins by occupying all neighbors of some vertex. The MBTD game has a hypergraph interpretation via $v$7. It is known that every tree lies in $v$8, that there are infinitely many connected cubic graphs in $v$9, that no minimum degree condition guarantees Dominator’s win when Staller starts, and that the problem is PSPACE-complete on split and bipartite graphs (Gledel et al., 2019). In the cubic setting, every connected cubic bipartite graph lies in $o(G)\in\{D,N,S\}$0, whereas generalized Petersen graphs $o(G)\in\{D,N,S\}$1 lie in $o(G)\in\{D,N,S\}$2 for every $o(G)\in\{D,N,S\}$3 (Forcan et al., 2020).

Biased variants allow Dominator and Staller to claim unequal numbers of vertices per round. In the $o(G)\in\{D,N,S\}$4-biased Maker-Breaker domination game, threshold parameters $o(G)\in\{D,N,S\}$5, $o(G)\in\{D,N,S\}$6, $o(G)\in\{D,N,S\}$7, and $o(G)\in\{D,N,S\}$8 measure the minimum bias required for one side to force a win. The $o(G)\in\{D,N,S\}$9-local domination number $D$0 gives upper bounds on $D$1 and $D$2; in particular, for any claw-free graph $D$3, $D$4. On trees, the star-partition width $D$5 satisfies $D$6 for any nontrivial graph $D$7, with equality for trees (Brešar et al., 14 Mar 2025). A different biased model, in which Dominator may claim up to $D$8 vertices each round and Staller exactly one, yields a full characterization of all trees on which Dominator has a winning strategy and exact round counts on powers of paths and cycles (Bagdas et al., 2024).

5. Criticality, predomination, and fast-winning theory

Criticality studies how sensitively optimal play depends on edge deletion or, in some formulations, edge addition. For the S-game parameter $D$9, a graph is $S$0-$S$1-critical if $S$2 and $S$3 for every edge $S$4. Connected bipartite $S$5-$S$6-critical graphs are exactly the graphs in a family $S$7, and connected graphs with a cut-vertex that are $S$8-$S$9-critical are exactly the graphs in two infinite families $G$00 and $G$01. Among connected triangle-free graphs, $G$02 is the only non-bipartite $G$03-$G$04-critical graph (Brešar et al., 23 Jul 2025).

Related work treats “critical” graphs for the four MBD numbers $G$05, $G$06, $G$07, and $G$08. A connected graph is $G$09-$G$10-critical if and only if it is a nontrivial star $G$11. A connected graph is $G$12-$G$13-critical if and only if it is $G$14 with $G$15. Connected bipartite $G$16-$G$17-critical graphs are exactly $G$18 with $G$19 and the bipartite family $G$20 in which exactly two vertices in each part are bipartite dominating and all others have degree $G$21 (Divakaran et al., 2024).

Predomination adds an initial dominated set $G$22. A predominated graph $G$23 is MBD-critical if Staller wins on $G$24, but Dominator wins on $G$25 for every $G$26. On trees, Staller wins on $G$27 if and only if there exists $G$28 such that $G$29 is a substructure of $G$30 and $G$31. Moreover, $G$32 is MBD-critical if and only if there is a substructure $G$33 in $G$34 with

$G$35

There is a linear-time algorithm for deciding whether a given predominated tree is MBD-critical, and a large class of MBD-critical predominated cacti is also known (Bujtás et al., 14 Mar 2025).

Fast-winning theory complements Dominator’s parameters by analyzing the minimum number of Staller moves needed to force a win. For odd paths,

$G$36

while for even paths both parameters are $G$37. For tadpole graphs $G$38, Staller has a winning S-game if and only if $G$39 is even and $G$40 is odd, and then

$G$41

On trees, the classes $G$42 characterize precisely those trees $G$43 with $G$44. For subdivided stars $G$45 with at least two odd branch lengths, if $G$46 and $G$47 are the two smallest such numbers, then

$G$48

For caterpillars, exact formulas are available for both $G$49 and $G$50 (Bujtás et al., 2022, Bujtás et al., 2022).

6. Complexity and algorithmic landscape

The winner-determination problem for the standard Maker-Breaker domination game is PSPACE-complete in general, and remains PSPACE-complete even on bipartite graphs and split graphs (Duchêne et al., 2018). Later work records PSPACE-completeness on chordal graphs as well, while also emphasizing that the problem is polynomial on cographs, trees, outerplanar graphs, and block graphs, and is in NP for interval graphs (Bagan et al., 2024).

Beyond classical complexity, the short-move versions of the game exhibit a Maker-versus-Breaker split in the $G$51-hierarchy. Deciding whether Staller can isolate a vertex in at most $G$52 moves is W[1]-complete, while deciding whether Dominator can dominate the graph in at most $G$53 moves is W[2]-complete. These results are obtained by transferring known and new hardness results for general Maker-Breaker games on hypergraphs to the closed-neighborhood hypergraph of a graph (Bagan et al., 13 Jan 2026).

At the same time, several structural parameters render the game fixed-parameter tractable. The Maker-Breaker domination game is FPT when parameterized by neighborhood diversity, modular width, $G$54-fewness, distance to cluster, and feedback edge number. The corresponding algorithms combine kernelization, modular decomposition, primeval decomposition, and path-shortening arguments. For neighborhood diversity there is a linear kernel of size at most $G$55, and for modular width the runtime is $G$56 with $G$57 (Bagan et al., 13 Jan 2026).

Algorithmic exactness is especially strong on trees and related sparse classes. Residual-graph methods determine $G$58 and $G$59 for arbitrary trees (Gledel et al., 2018); substructure methods characterize Staller-win trees and support polynomial-time computation of $G$60 and $G$61 in several tree classes (Bujtás et al., 2022). For predominated trees, MBD-criticality can be recognized in linear time (Bujtás et al., 14 Mar 2025).

7. Open problems and current directions

Several open problems organize the present frontier. In the disjoint domination game, it remains open to characterize the maker-win graphs in the maker-start non-biased game and to analyze the $G$62-biased game for $G$63 (Bujtás et al., 2014). In the biased standard game, extending the threshold analysis beyond the $G$64 regime and determining the exact value of $G$65 on grids remain explicit problems (Brešar et al., 14 Mar 2025).

For criticality, higher-move analogues are largely unresolved. A full description is known for many $G$66- and $G$67-move critical graphs, but the structure of $G$68-critical graphs for $G$69 remains widely open (Divakaran et al., 2024). In the predominated setting, Problem 5.4 asks whether the cactus family $G$70 captures all atomic MBD-critical cacti (Bujtás et al., 14 Mar 2025).

Total-domination variants pose parallel structural questions. Open problems include characterizing which connected cubic graphs lie in $G$71 versus $G$72, deciding whether large girth plus regularity forces Dominator’s win in the D-game on $G$73-regular graphs with $G$74, and determining $G$75 and $G$76 for standard graph classes such as trees, grids, and bipartite graphs (Gledel et al., 2019, Divakaran et al., 23 Jul 2025).

A broad pattern across the literature is the interaction between local neighborhood obstructions and global decomposition structure. Pairing dominating sets, perfect $G$77-factors, residual graphs, star partitions, substructure families, and hypergraph transversals repeatedly serve as the mechanisms that convert graph structure into optimal-play theorems. This suggests that further progress is likely to come from increasingly precise correspondences between domination-theoretic decompositions and positional-game strategy classes (Duchêne et al., 2018, Bagan et al., 2024, Bujtás et al., 14 Mar 2025).