2-Movable Total Domination in Graphs

• 2-movable total domination is a refined concept where a set of vertices not only totally dominates a graph but also allows any pair of its vertices to be replaced while preserving domination.
• For join graphs with both parts nontrivial, the invariant reaches the minimum value of 2, showcasing the power of complete bipartite connectivity in ensuring domination.
• In corona graphs, the 2-movable total domination number scales as the product of the base graph order and the total domination number of the attached graph, reflecting local domination constraints.

A 2-movable total dominating set is a non-empty vertex subset $T\subseteq V(G)$ of a connected graph $G$ such that $T$ is a total dominating set and every pair of vertices in $T$ can either be deleted without destroying total domination, or replaced by two vertices outside $T$ adjacent to the deleted vertices so that total domination is preserved. The associated invariant, the 2-movable total domination number $\gamma_{mt}^{2}(G)$, is the minimum cardinality of such a set. The paper "On 2-Movable Total Domination in the Join and Corona of Graphs" develops this notion for join and corona constructions and derives explicit formulas for these graph operations (Pedrano et al., 14 Aug 2025).

1. Definition and formal conditions

Let $G$ be a connected graph. A non-empty set $T\subseteq V(G)$ is a 2-movable total dominating set of $G$ if two conditions hold (Pedrano et al., 14 Aug 2025).

First, $T$ must be a total dominating set, meaning that every vertex of $G$0 is adjacent to some vertex of $G$1. In the notation used in the source,

$G$2

Second, for every pair $G$3, at least one of the following alternatives must hold (Pedrano et al., 14 Aug 2025):

• $G$4 is a total dominating set in $G$5; or
• there exist $G$6 such that $G$7 is adjacent to $G$8, $G$9 is adjacent to $T$0, and

$T$1

is a total dominating set in $T$2.

The definition therefore combines ordinary total domination with a pairwise replacement condition. The source explicitly notes that every 2-movable total dominating set is a total dominating set, but that the new notion is more restrictive than usual total domination and also than 2-movable domination without the total-domination requirement (Pedrano et al., 14 Aug 2025).

2. The invariant $T$3

The 2-movable total domination number of $T$4, denoted $T$5, is defined by

$T$6

Any 2-movable total dominating set of size $T$7 is called a $T$8-set of $T$9 (Pedrano et al., 14 Aug 2025).

The paper records a basic lower bound: for any connected graph $T$0 with order $T$1,

$T$2

(Pedrano et al., 14 Aug 2025). This lower bound is consistent with the pair-removal condition built into the definition: the structure is designed to control what happens when two vertices of the chosen set are removed.

The paper also gives a comparison with the corresponding non-total parameter: $T$3 (Pedrano et al., 14 Aug 2025). This inequality formalizes the fact that the total-domination constraint can only make the minimization problem at least as restrictive.

3. Join graphs

For graphs $T$4 and $T$5 of order at least $T$6, the paper proves the exact formula

$T$7

(Pedrano et al., 14 Aug 2025).

The construction is direct. One takes a set $T$8 with $T$9 and $T$0. In the join $T$1, every vertex of $T$2 is adjacent to every vertex of $T$3, so every vertex is adjacent to one of the two selected vertices. The proof outline further states that replacing $T$4 and $T$5 with other vertices in the corresponding parts yields another 2-element total dominating set, which establishes the required 2-movability condition (Pedrano et al., 14 Aug 2025).

A concrete instance is given by $T$6 and $T$7. In $T$8, two vertices, one from each part, dominate the whole graph and can be moved in the sense of the definition (Pedrano et al., 14 Aug 2025).

The paper isolates a special case when one summand is $T$9. If $\gamma_{mt}^{2}(G)$0, then

$\gamma_{mt}^{2}(G)$1

where $\gamma_{mt}^{2}(G)$2 is the ordinary total domination number of $\gamma_{mt}^{2}(G)$3 (Pedrano et al., 14 Aug 2025). The proof sketch states that any minimal total dominating set of $\gamma_{mt}^{2}(G)$4 remains a minimal 2-movable total dominating set for $\gamma_{mt}^{2}(G)$5, and that smaller choices lead to contradiction.

This pair of results shows that the join behaves in two sharply different ways depending on whether both parts have order at least $\gamma_{mt}^{2}(G)$6 or one part is a single vertex. A plausible implication is that the full bipartite adjacency created by a nontrivial join is strong enough to force the parameter down to its minimum possible value $\gamma_{mt}^{2}(G)$7, whereas adjoining a single universal vertex preserves the total-domination threshold of the original graph.

4. Corona graphs

The paper also studies the corona $\gamma_{mt}^{2}(G)$8, formed by taking one copy of $\gamma_{mt}^{2}(G)$9 and, for each vertex $G$0, attaching a copy $G$1 of $G$2 with edges from $G$3 to all vertices in $G$4 (Pedrano et al., 14 Aug 2025).

For connected graphs $G$5 and $G$6, the main formula is the following: if $G$7 and $G$8, then

$G$9

(Pedrano et al., 14 Aug 2025).

The construction described in the source is local across the attached copies. For each $T\subseteq V(G)$0, choose a $T\subseteq V(G)$1-set $T\subseteq V(G)$2 in the copy $T\subseteq V(G)$3, and let

$T\subseteq V(G)$4

The paper states that this union is a minimal 2-movable total dominating set (Pedrano et al., 14 Aug 2025).

The proof outline emphasizes why fewer vertices are not sufficient: if one attempted to use fewer selected vertices, then at least one attached copy $T\subseteq V(G)$5 would receive too few selected vertices, violating the minimum required by $T\subseteq V(G)$6. The same outline states that the structure of the corona ensures that total domination acts locally in each $T\subseteq V(G)$7, while 2-movability can be handled by swapping in vertices from these sets (Pedrano et al., 14 Aug 2025).

The example $T\subseteq V(G)$8 and $T\subseteq V(G)$9 yields

$G$0

The source realizes this value by taking both vertices in each copy of $G$1 (Pedrano et al., 14 Aug 2025).

5. Explicit formulas and worked instances

The main formulas recorded in the source can be organized as follows.

Graph operation	Hypotheses	Value of $G$2
$G$3	$G$4	$G$5
$G$6	$G$7	$G$8
$G$9	$T$0, $T$1	$T$2

These formulas capture the principal outcomes of the paper (Pedrano et al., 14 Aug 2025). The join case gives an absolute minimum of $T$3 once both factors are nontrivial. The corona case instead scales with $T$4 and the total domination number of the attached graph $T$5.

The worked examples in the source illustrate these behaviors concretely. For $T$6, one vertex from each part suffices. For $T$7, every copy of $T$8 contributes its full total dominating set, leading to a total of $T$9 selected vertices (Pedrano et al., 14 Aug 2025).

A common misunderstanding is to treat these formulas as consequences of ordinary domination. The paper’s statements are specifically about total domination together with the pairwise movability condition. The distinction matters, because the source explicitly records that 2-movable total domination is more restrictive than both ordinary total domination and 2-movable domination without the total condition (Pedrano et al., 14 Aug 2025).

6. Placement within total domination and reconfiguration literature

The underlying notion of a total dominating set is standard in the cited literature: a set $G$00 such that every vertex in $G$01 has a neighbor in $G$02 (Akbari et al., 2015). This broader context includes several adjacent invariants.

One direction concerns total domatic number. The paper "Cubic Graphs with Total Domatic Number at Least Two" defines the total domatic number $G$03 as the maximum number of total dominating sets that partition the vertex set, and characterizes cubic graphs whose vertex sets can be partitioned into two total dominating sets by the absence of a specific forbidden subgraph $G$04 (Akbari et al., 2015). By definition, this is a partition problem, whereas $G$05 is a minimum-cardinality problem for a single set satisfying a pairwise replacement property.

Another direction concerns reconfiguration. The paper "On $G$06-Total Dominating Graphs" defines the graph $G$07, whose vertices correspond to total dominating sets of cardinality at most $G$08, with adjacency given by adding or deleting a single vertex (Alikhani et al., 2017). The same source uses $G$09 to study when one total dominating set can be transformed into another through intermediate total dominating sets. It also introduces the threshold $G$10, the smallest integer $G$11 such that $G$12 is connected for all $G$13 (Alikhani et al., 2017).

This reconfiguration viewpoint is related but not identical to the 2-movable total domination number. The source on $G$14-total dominating graphs states that, in that setting, the term 2-movable refers to the ability to reconfigure between any two MTDSs using single additions or deletions, and it derives exact values of $G$15 and $G$16, including the exceptional behavior of $G$17 (Alikhani et al., 2017). A plausible implication is that 2-movability in total domination supports more than one formalization in the literature: one based on local replacement inside a single total dominating set, and another based on connectivity in a reconfiguration graph.

Within that landscape, the contribution of (Pedrano et al., 14 Aug 2025) is to give exact formulas for $G$18 under two graph operations. The results identify graph families where the parameter is either forced down to $G$19, as in nontrivial joins, or determined by a product formula involving $G$20, as in the stated corona regime.