Maker-Breaker Total Domination Number

• Maker-Breaker total domination number is a game-theoretic invariant that measures the fewest moves needed by Dominator to form a total dominating set in a graph.
• It refines conventional domination concepts by integrating game dynamics, relating classic total domination with a competitive move-count framework.
• The parameter features sharp universal bounds, explicit realization theorems, and complexity results that clarify strategic outcomes across various graph families.

The Maker-Breaker total domination number is a game-theoretic invariant attached to the Maker-Breaker total domination game on a graph $G$. In that game, Dominator and Staller alternately select previously unplayed vertices; Dominator wins if the vertices he has chosen contain a total dominating set of $G$, while Staller wins if she isolates a vertex by claiming all of its neighbors. The parameter $\gamma_{\rm MBT}(G)$ is the minimum number of Dominator’s moves needed to force a win when Dominator starts, and $\gamma'_{\rm MBT}(G)$ is defined analogously when Staller starts; if Dominator has no winning strategy, the corresponding value is set to $\infty$ (Divakaran et al., 23 Jul 2025).

1. Formal game model and basic terminology

A set $D\subseteq V(G)$ is a total dominating set if every vertex of $G$ has a neighbor in $D$, equivalently

$\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$

The Maker-Breaker total domination game is played on a graph $G$ by two players who alternately choose previously unplayed vertices. Dominator’s objective is to build a total dominating set using only his claimed vertices. Staller’s objective is to prevent this by claiming all neighbors of some vertex; when she does so, she isolates that vertex and wins. The literature distinguishes the D-game, in which Dominator starts, from the S-game, in which Staller starts. Before the introduction of the numerical parameters $G$0 and $G$1, the game was studied through outcome classes: $G$2 if Dominator wins regardless of who starts, $G$3 if Staller wins regardless of who starts, and $G$4 if the first player wins (Gledel et al., 2019).

The numerical parameters refine that outcome classification by measuring the minimum number of Dominator’s moves in a winning strategy. They are therefore defined only conditionally on the existence of such a strategy; otherwise they are assigned the value $G$5. This places the Maker-Breaker total domination number in the same general category as fast-winning invariants in positional graph games, but with the total domination constraint rather than ordinary domination (Divakaran et al., 23 Jul 2025).

2. Relation to total domination, Maker-Breaker domination, and other game parameters

The parameter is anchored in three adjacent notions. First, it is constrained by the ordinary total domination number $G$6, since every winning set for Dominator in the Maker-Breaker total domination game must itself be a total dominating set. Second, it is related to the Maker-Breaker domination game, where Dominator only needs a dominating set rather than a total dominating set. Third, it should be distinguished from the game total domination number $G$7, which belongs to a different game model in which Dominator and Staller alternately choose legal vertices and both players’ moves contribute to a single growing total dominating set (Divakaran et al., 23 Jul 2025).

The 2019 MBTD paper makes the comparison with Maker-Breaker domination explicit: if Staller wins the Maker-Breaker domination game on $G$8, then Staller also wins the Maker-Breaker total domination game on $G$9; equivalently, if Dominator wins the Maker-Breaker total domination game on $\gamma_{\rm MBT}(G)$0, then Dominator also wins the Maker-Breaker domination game on $\gamma_{\rm MBT}(G)$1. In this precise sense, the total-domination version is stronger for Staller and harder for Dominator (Gledel et al., 2019).

The contrast with $\gamma_{\rm MBT}(G)$2 is structural rather than merely terminological. In the total domination game, a move is legal only if it strictly increases the set of totally dominated vertices, and $\gamma_{\rm MBT}(G)$3 counts the total number of chosen vertices when Dominator starts and both players play optimally. That game is equivalent to a transversal game on the open neighborhood hypergraph $\gamma_{\rm MBT}(G)$4. The Maker-Breaker total domination number, by contrast, counts only Dominator’s moves in a disjoint-claim positional game with an explicit Maker-Breaker win condition (Bujtás, 2017).

3. Universal bounds and sharpness

The foundational inequalities established for the new parameters are

$\gamma_{\rm MBT}(G)$5

and, in the formulation emphasized in the 2025 paper,

$\gamma_{\rm MBT}(G)$6

The first inequality reflects the fact that Dominator’s claimed set at the moment of victory is a total dominating set. The second follows from the observation that a D-game can be viewed as an S-game in which Staller skips her first move, together with the No-Skip Lemma. The same paper proves the universal upper bounds

$\gamma_{\rm MBT}(G)$7

whenever the corresponding parameter is finite, and it proves that both bounds are sharp (Divakaran et al., 23 Jul 2025).

Sharpness is demonstrated by explicit connected constructions. For even $\gamma_{\rm MBT}(G)$8, one construction takes a path $\gamma_{\rm MBT}(G)$9 and attaches a copy of $\gamma'_{\rm MBT}(G)$0 to each vertex, yielding a graph $\gamma'_{\rm MBT}(G)$1 with

$\gamma'_{\rm MBT}(G)$2

For odd $\gamma'_{\rm MBT}(G)$3, the first $\gamma'_{\rm MBT}(G)$4-attachment is replaced by a special “twin $\gamma'_{\rm MBT}(G)$5” gadget, producing a graph $\gamma'_{\rm MBT}(G)$6 with $\gamma'_{\rm MBT}(G)$7. Analogous parity-sensitive constructions establish sharpness for the Staller-start bound as well, using $\gamma'_{\rm MBT}(G)$8 in the even case and a modified graph $\gamma'_{\rm MBT}(G)$9 in the odd case (Divakaran et al., 23 Jul 2025).

The same paper also exhibits connected graphs for which the game-theoretic and classical invariants coincide. In particular, it gives connected graphs satisfying

$\infty$0

The underlying constructions are built from a clique $\infty$1, together with $\infty$2 copies of $\infty$3, and one or two additional vertices arranged so that Dominator can realize the required total dominating pattern in exactly $\infty$4 moves (Divakaran et al., 23 Jul 2025).

4. Realization theorems and separation phenomena

A central contribution of the 2025 paper is a set of realization theorems showing that the Maker-Breaker total domination numbers can be prescribed largely independently of the corresponding Maker-Breaker domination numbers. For every pair of integers $\infty$5 with $\infty$6, there exists a connected graph $\infty$7 such that

$\infty$8

Likewise, there exists a connected graph $\infty$9 such that

$D\subseteq V(G)$0

and there exists a connected graph $D\subseteq V(G)$1 such that

$D\subseteq V(G)$2

These are exact existence results, not asymptotic statements (Divakaran et al., 23 Jul 2025).

The constructions are explicit and use families of triangles together with a small number of specially connected vertices denoted $D\subseteq V(G)$3, $D\subseteq V(G)$4, and $D\subseteq V(G)$5. For $D\subseteq V(G)$6 and $D\subseteq V(G)$7, the graphs $D\subseteq V(G)$8 are built from either $D\subseteq V(G)$9 triangles when $G$0, or from $G$1 triangles when $G$2, together with two special vertices $G$3. For the Staller-start realization $G$4, $G$5, the graphs $G$6 use three special vertices $G$7. For the mixed realization $G$8, $G$9, the graphs $D$0 are built from $D$1 triangles and two special vertices $D$2 (Divakaran et al., 23 Jul 2025).

These realization theorems imply that the gap between ordinary Maker-Breaker domination parameters and total-domination Maker-Breaker parameters can be made arbitrarily large, even on connected graphs. They also show that the D-game and S-game total domination numbers are not rigidly coupled beyond the inequality $D$3. The paper summarizes this by stating that the MBTD parameters are highly flexible and can be independently tuned over all feasible integer pairs (Divakaran et al., 23 Jul 2025).

5. Outcome classifications and graph families relevant to finiteness

Because $D$4 and $D$5 are finite only when Dominator has an appropriate winning strategy, outcome classifications from the earlier MBTD literature determine large domains where these parameters are finite or infinite. The 2019 paper gives a complete classification on cycles: $D$6 It also proves that for $D$7, the game on the grid $D$8 is $D$9 if both $\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$0 and $\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$1 are even, and $\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$2 otherwise; for $\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$3 and $\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$4, the prism-type graph $\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$5 is $\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$6. Cacti are classified by a structural theorem: a cactus with at least two blocks is $\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$7 exactly when its vertex set can be partitioned into $\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$8-sets each inducing a $\forall v\in V(G),\quad N(v)\cap D\neq \emptyset.$9; it is $G$0 exactly when iterative removal of end-blocks $G$1 yields an $G$2-star cactus; otherwise it is $G$3. For trees, the corollary is especially sharp: the MBTD game is $G$4 if $G$5 with $G$6, and $G$7 otherwise. The same paper also proves that there are infinitely many connected cubic graphs in which Staller wins and that no minimum degree condition is sufficient to guarantee that Dominator wins when Staller starts (Gledel et al., 2019).

The 2020 study of connected cubic graphs refines this picture within the cubic setting. If a connected cubic graph on $G$8 vertices contains a diamond-factor, then it belongs to $G$9. If it contains a triangle-factor, then it is in $G$00 only for $G$01, and otherwise it is in $G$02. For every $G$03, the generalized Petersen graph $G$04 lies in $G$05, whereas every cubic bipartite graph lies in $G$06. For connected cubic graphs with a claw-factor, the classification depends on the number $G$07 of vertex-disjoint claws: $G$08 implies $G$09, while $G$10 implies $G$11 (Forcan et al., 2020).

Combined with the convention that $G$12 when Dominator has no winning strategy, these outcome results delimit many graph classes on which the Maker-Breaker total domination number is finite and many on which it is forced to be infinite. This suggests that structural winner classification remains a prerequisite for systematic computation of $G$13 outside specially controlled families.

6. Hypergraph interpretation, complexity, and placement in the literature

The Maker-Breaker total domination game admits a direct hypergraph interpretation. Given a graph $G$14, one considers the hypergraph $G$15 with vertex set $G$16 and hyperedges

$G$17

that is, the family of open neighborhoods. In this formulation, Maker corresponds to Staller and Breaker to Dominator. Staller wins by fully claiming one hyperedge $G$18, which is exactly the event of isolating $G$19, while Dominator wins by preventing every such event. The same 2019 paper proves that deciding the outcome of the MBTD game is PSPACE-complete on split graphs and, by a modified reduction, also PSPACE-complete on bipartite graphs (Gledel et al., 2019).

This open-neighborhood hypergraph viewpoint is closely aligned with the hypergraph formulation of the total domination game. For an isolate-free graph $G$20, the total domination game is equivalent to the transversal game on the open neighborhood hypergraph $G$21, with

$G$22

The shared use of open neighborhoods is conceptually important: both theories encode total domination through the same neighborhood system, but they impose different adversarial objectives and measure different quantities. In $G$23, the quantity of interest is the total number of played vertices under legal-move dynamics; in $G$24, it is the minimum number of Dominator’s moves in a Maker-Breaker win (Bujtás, 2017).

The broader complexity and parameterized-complexity program has, so far, been developed for the ordinary Maker-Breaker domination game rather than its total-domination variant. The 2026 paper on parameterized complexity explicitly states that it is not about total domination and does not define a Maker-Breaker total domination number, although it develops hypergraph, move-count, and module-reduction methods that are conceptually close to what a total-domination extension would require (Bagan et al., 13 Jan 2026). A plausible implication is that a full algorithmic theory of $G$25 remains less developed than the corresponding theory for ordinary Maker-Breaker domination.

In this sense, the Maker-Breaker total domination number occupies a precise position within the modern theory of domination-type games. It refines the 2019 winner-class framework for the MBTD game by introducing move-count invariants; it is bounded below by the classical total domination number and constrained by start-player asymmetry; it differs fundamentally from the game total domination number; and it already supports sharp universal bounds, coincidence constructions, and complete pair-realization theorems (Divakaran et al., 23 Jul 2025).