Staller-Start Maker-Breaker Total Domination

• Staller-Start Maker-Breaker total domination is a parameter that measures the minimum number of Dominator moves needed to force a win in a total domination game when Staller starts.
• It refines outcome classifications by providing explicit sharp bounds and construction theorems, demonstrating precise move-count strategies on various graph families.
• The framework differentiates itself from classical total domination games through its focus on efficiency, strategic block construction, and separation phenomena among competing game invariants.

The Staller-start Maker-Breaker total domination number of a graph $G$, denoted $\gamma_{\rm MBT}'(G)$, is the minimum number of moves of Dominator needed to force a win in the Maker-Breaker total domination game when Staller starts, provided that Dominator has a winning strategy; if Dominator cannot win the Staller-start game, then $\gamma_{\rm MBT}'(G)=\infty$ (Divakaran et al., 23 Jul 2025). As a quantitative refinement of the outcome classes $\mathcal{D}$, $\mathcal{S}$, and $\mathcal{N}$, the parameter measures not merely whether second-player Dominator can still prevail, but how rapidly he can complete a total dominating set against optimal interference.

1. Formal definition and game-theoretic setting

In the modern formulation of the Maker-Breaker total domination game, the board is the vertex set of a finite simple graph $G=(V,E)$. Dominator plays as Maker and wins as soon as the set of vertices he has chosen forms a total dominating set of $G$. Staller plays as Breaker and wins if the game ends with all vertices chosen and Dominator has never achieved a total dominating set (Divakaran et al., 23 Jul 2025). A total dominating set $S \subseteq V(G)$ is a set such that every vertex of $G$ has a neighbor in $\gamma_{\rm MBT}'(G)$0.

Two starting conventions are standard. In the D-game, Dominator starts; in the S-game, Staller starts. The corresponding move-count parameters are $\gamma_{\rm MBT}'(G)$1 and $\gamma_{\rm MBT}'(G)$2. Both count Dominator’s moves, not total turns. Thus $\gamma_{\rm MBT}'(G)$3 is a second-player efficiency parameter: it records the minimum number of Dominator moves needed to force a total dominating set after Staller has already occupied one vertex (Divakaran et al., 23 Jul 2025).

An important terminological complication is that earlier work on the same underlying game used the opposite Maker-Breaker naming convention: in the 2020 cubic-graph study, Staller plays the role of Maker and wins by claiming an open neighborhood $\gamma_{\rm MBT}'(G)$4, while Dominator plays the role of Breaker and wins by claiming a total dominating set (Forcan et al., 2020). The win conditions are the same underlying complementary objectives, but the player-role labels differ across the literature.

2. Bounds, monotonicity, and extremal behavior

The basic order bounds are immediate from alternation. If $\gamma_{\rm MBT}'(G)$5 and $\gamma_{\rm MBT}'(G)$6, then

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

The corresponding D-game bound is

$\gamma_{\rm MBT}'(G)$8

Both bounds are sharp in the strongest sense: for every integer $\gamma_{\rm MBT}'(G)$9, there exist connected graphs attaining equality in the D-game and in the S-game respectively (Divakaran et al., 23 Jul 2025).

The parameter is monotone with respect to move order in the expected direction. The No-Skip Lemma implies

$\gamma_{\rm MBT}'(G)=\infty$0

so allowing Staller the first move cannot reduce the number of Dominator moves required for a forced win (Divakaran et al., 23 Jul 2025). The 2025 paper also states the comparison

$\gamma_{\rm MBT}'(G)=\infty$1

placing the Maker-Breaker total domination numbers above the classical total domination number and above the corresponding Maker-Breaker domination parameter in the sense recorded there (Divakaran et al., 23 Jul 2025).

The order upper bounds are realized by explicit connected constructions built from paths with attached $\gamma_{\rm MBT}'(G)=\infty$2-blocks and, in the odd cases, a special initial gadget. In the S-game sharpness construction $\gamma_{\rm MBT}'(G)=\infty$3, Dominator needs exactly three moves in the distinguished gadget $\gamma_{\rm MBT}'(G)=\infty$4 plus two moves for each remaining $\gamma_{\rm MBT}'(G)=\infty$5, yielding

$\gamma_{\rm MBT}'(G)=\infty$6

with $\gamma_{\rm MBT}'(G)=\infty$7 (Divakaran et al., 23 Jul 2025).

3. Realization theorems and separation phenomena

A central feature of $\gamma_{\rm MBT}'(G)=\infty$8 is its flexibility. The 2025 paper proves three realization theorems showing that the Staller-start parameter can be prescribed independently of several related invariants, subject only to the trivial order relation $\gamma_{\rm MBT}'(G)=\infty$9 (Divakaran et al., 23 Jul 2025).

Realized pair	Statement	Family
$\mathcal{D}$0	Exists for all $\mathcal{D}$1	$\mathcal{D}$2, $\mathcal{D}$3
$\mathcal{D}$4	Exists for all $\mathcal{D}$5	$\mathcal{D}$6, $\mathcal{D}$7
$\mathcal{D}$8	Exists for all $\mathcal{D}$9	$\mathcal{S}$0

For the specifically Staller-start realization, the graphs $\mathcal{S}$1 and $\mathcal{S}$2 show that $\mathcal{S}$3 can be arbitrarily larger than $\mathcal{S}$4. Likewise, the family $\mathcal{S}$5 gives

$\mathcal{S}$6

for every $\mathcal{S}$7, so the difference $\mathcal{S}$8 can be arbitrarily large (Divakaran et al., 23 Jul 2025).

The mechanism behind these constructions is uniform. The graphs are assembled from many triangles together with a small set of heavy vertices $\mathcal{S}$9. Dominator’s winning strategy typically begins by choosing one heavy vertex, thereby totally dominating most of the graph and reducing the remaining task to “hitting” one unresolved vertex in each of a controlled family of triangles. Staller’s optimal delaying strategy is to occupy the complementary heavy vertex as early as possible, forcing Dominator to spend one move per residual triangle. Consequently, the value of $\mathcal{N}$0 can be engineered by controlling the number of such blocks (Divakaran et al., 23 Jul 2025).

4. Prehistory in cubic graphs: outcome classifications before the numerical invariant

Before $\mathcal{N}$1 was introduced, the cubic-graph paper “Maker-Breaker total domination game on cubic graphs” studied only who wins the S-game and D-game under optimal play; it did not define a numerical Maker-Breaker total domination number (Forcan et al., 2020). Nevertheless, its structural classifications determine whether the later parameter $\mathcal{N}$2 is finite or infinite on several cubic families: if Dominator wins the S-game, then $\mathcal{N}$3; if Staller wins the S-game, then $\mathcal{N}$4. This implication follows from the later definition of $\mathcal{N}$5 as a move count conditioned on Dominator’s S-game win (Divakaran et al., 23 Jul 2025).

Cubic class	S-game winner in the 2020 classification	Consequence for $\mathcal{N}$6
Diamond-factor cubic graphs	Dominator	Finite
Triangle-factor, $\mathcal{N}$7	Dominator	Finite
Triangle-factor, $\mathcal{N}$8	Staller	$\mathcal{N}$9
$G=(V,E)$0, $G=(V,E)$1	Staller	$G=(V,E)$2
Cubic bipartite graphs	Dominator	Finite
Claw-factor, $G=(V,E)$3	Dominator	Finite
Claw-factor, $G=(V,E)$4	Staller	$G=(V,E)$5

Several results are especially notable. A cubic graph with a diamond-factor is $G=(V,E)$6, and every cubic bipartite graph is $G=(V,E)$7, so Dominator wins even when Staller starts (Forcan et al., 2020). By contrast, $G=(V,E)$8 is $G=(V,E)$9 for all $G$0, and triangle-factor cubic graphs with $G$1 are also $G$2, so on those classes the later parameter is necessarily infinite (Forcan et al., 2020).

These precursor results are structural rather than quantitative. They determine existence of a second-player Dominator win, but they do not optimize the number of Dominator moves. In that sense, the 2025 introduction of $G$3 converts an outcome theory into a genuine counting theory.

5. Distinction from the Staller-start game total domination number

The Staller-start Maker-Breaker total domination number should not be confused with the Staller-start game total domination number $G$4 from the total domination game literature (Henning et al., 2016). The two settings use similar graph-theoretic objects but different game semantics.

In the total domination game, Dominator and Staller alternately select legal vertices, each move must strictly increase the number of vertices already totally dominated, the process always terminates in a total dominating set, Dominator tries to minimize the total number of played vertices, and Staller tries to maximize it. The Staller-start parameter there is

$G$5

the total length of the game under optimal play when Staller starts (Henning et al., 2016). For every isolate-free graph,

$G$6

and for forests one has

$G$7

(Henning et al., 2016). A general upper bound of Bujtás gives

$G$8

for graphs on $G$9 vertices in which every component contains at least three vertices (Bujtás, 2017).

By contrast, $S \subseteq V(G)$0 belongs to a win/lose Maker-Breaker game. It counts only Dominator’s moves, and it may take the value $S \subseteq V(G)$1 if Staller can prevent a second-player Dominator win (Divakaran et al., 23 Jul 2025). The two parameters therefore answer different questions: $S \subseteq V(G)$2 measures how long an unavoidable total domination process lasts, whereas $S \subseteq V(G)$3 measures how efficiently Dominator can still force total domination when the opponent is attempting to block it.

A related misconception concerns the phrase “Staller-start.” In $S \subseteq V(G)$4, Staller starts, but the parameter still quantifies Dominator’s winning speed. It is not a measure of how quickly Staller wins.

6. Methods, related frameworks, and open directions

The 2025 development of $S \subseteq V(G)$5 relies on a small collection of recurring strategy types. One is the No-Skip Lemma, which converts first-move information into inequalities between D-game and S-game parameters. Another is the use of heavy vertices $S \subseteq V(G)$6 with large neighborhoods, allowing Dominator to compress the unresolved part of the game into independent local blocks, usually triangles or $S \subseteq V(G)$7-gadgets. A third is the construction of explicit delaying strategies for Staller that force occupation of all heavy vertices and then require one additional Dominator move per unresolved block (Divakaran et al., 23 Jul 2025).

The paper leaves several standard families unresolved. It does not systematically compute $S \subseteq V(G)$8 or $S \subseteq V(G)$9 for paths $G$0, cycles $G$1 beyond outcome information, complete graphs $G$2, complete bipartite graphs $G$3, or trees (Divakaran et al., 23 Jul 2025). It also does not settle the complexity of deciding whether $G$4 for a given graph $G$5 and integer $G$6 (Divakaran et al., 23 Jul 2025).

A plausible methodological implication is that techniques from the ordinary Maker-Breaker domination game on Cartesian products may be adaptable to the total version. In that setting, generalized pairing strategies, edge-deletion monotonicity, and nontrivial path-cover decompositions yield exact Staller-start parameters on several products and prove that $G$7 whenever both factors admit nontrivial path covers (Dokyeesun, 2023). Since those arguments are structural rather than metric-specific, they suggest a route toward systematic computations of $G$8 on Cartesian products, although that extension is not carried out there.

Within the present literature, the Staller-start Maker-Breaker total domination number marks the transition from qualitative outcome theory to quantitative positional analysis. Earlier work classified when second-player Dominator can survive; the 2025 framework asks how many Dominator moves survival actually costs.