2-Movable Dominating Set in Graphs

• 2-movable dominating sets are defined as dominating sets that remain dominating after any two vertices are removed, or can be repaired by replacing them with adjacent outside vertices.
• This concept extends traditional domination notions by introducing a strict local replacement condition, establishing a framework for enhanced fault tolerance in graphs.
• Exact formulas for graph operations like joins and coronas highlight how 2-movability scales and distinguishes between global coverage and localized structure.

Searching arXiv for papers directly relevant to 2-movable domination and closely related domination variants. I’ll look up the specific arXiv records to ground the article in the cited literature. A 2-movable dominating set is a domination-theoretic structure that strengthens ordinary domination by imposing a local replacement condition on every pair of selected vertices. For a connected graph $G$, a non-empty set $S\subseteq V(G)$ is a 2-movable dominating set if $S$ is a dominating set and, for every pair $x,y\in S$, either $S\setminus\{x,y\}$ remains a dominating set, or there exist $u,v\in V(G)\setminus S$ adjacent to $x$ and $y$, respectively, such that $(S\setminus\{x,y\})\cup\{u,v\}$ is again a dominating set (Pedrano et al., 20 Aug 2025). The associated minimum cardinality is the 2-movable domination number, denoted $\gamma_m^2(G)$, and a minimum such set is a $S\subseteq V(G)$0-set (Pedrano et al., 20 Aug 2025). In the literature summarized here, the concept is introduced as a new domination variant and analyzed exactly for the join and corona of graphs, while related work situates it within broader studies of domination reconfiguration and other robust domination frameworks (Pedrano et al., 20 Aug 2025).

1. Formal definition and parameterization

The underlying setting is that of connected, finite, simple, undirected graphs (Pedrano et al., 20 Aug 2025). Ordinary domination requires that every vertex outside a set $S\subseteq V(G)$1 be adjacent to at least one vertex of $S\subseteq V(G)$2. The 2-movable condition adds a uniform two-vertex resilience requirement: after deleting any two vertices of $S\subseteq V(G)$3, domination must either persist immediately or be restorable by inserting two outside vertices, one adjacent to each removed vertex (Pedrano et al., 20 Aug 2025).

Formally, if $S\subseteq V(G)$4 is connected, then a non-empty $S\subseteq V(G)$5 is a 2-movable dominating set if:

1. $S\subseteq V(G)$6 is a dominating set, and
2. for every pair $S\subseteq V(G)$7, either

$S\subseteq V(G)$8

is a dominating set in $S\subseteq V(G)$9, or there exist $S$0 such that $S$1 is adjacent to $S$2, $S$3 is adjacent to $S$4, and

$S$5

is a dominating set in $S$6 (Pedrano et al., 20 Aug 2025).

The minimum cardinality of a 2-movable dominating set is the 2-movable domination number $S$7, and a minimum 2-movable dominating set is called a $S$8-set (Pedrano et al., 20 Aug 2025). In the paper’s explanatory language, the condition means that a dominating set can survive the deletion of any two of its vertices either without repair or by a two-vertex replacement drawn from outside the set (Pedrano et al., 20 Aug 2025).

This definition places emphasis on local substitutability rather than only global coverage. A plausible implication is that the parameter captures a restricted form of fault tolerance inside dominating configurations, but the formal development in the cited work is confined to the exact replacement condition above.

2. Position within domination theory

The concept is introduced as a new domination variant, but it is explicitly tied to earlier notions of domination and movable domination (Pedrano et al., 20 Aug 2025). The paper recalls ordinary domination and 1-movable domination, and notes that 2-movable domination is more restrictive than ordinary domination (Pedrano et al., 20 Aug 2025).

For any connected graph $S$9 of order $x,y\in S$0, the paper states the basic lower bound

$x,y\in S$1

(Pedrano et al., 20 Aug 2025). It also proves that 2-movable domination is at least as restrictive as ordinary domination and 1-movable domination, summarized in the hierarchy

$x,y\in S$2

in the intended interpretation reported in the paper summary (Pedrano et al., 20 Aug 2025).

The article on 2-movable total domination gives an analogous comparison in the total-domination setting: if $x,y\in S$3 is a 2-movable total dominating set of $x,y\in S$4, then $x,y\in S$5 is also a 2-movable dominating set, so

$x,y\in S$6

(Pedrano et al., 14 Aug 2025). This comparison does not redefine 2-movable domination, but it clarifies that the ordinary-domination version is the weaker of the two “movable” parameters.

This positioning matters because the 2-movable condition is not merely a cardinality refinement of domination. It encodes a specific local repair property that is stronger than being dominating and weaker than requiring total domination. The relationship to 1-movable domination also indicates that the passage from one removable vertex to arbitrary removable pairs is structurally nontrivial (Pedrano et al., 20 Aug 2025).

3. Exact value for the join of graphs

One of the main exact results concerns the join $x,y\in S$7, formed from disjoint copies of $x,y\in S$8 and $x,y\in S$9 by adding all edges between every vertex of $S\setminus\{x,y\}$0 and every vertex of $S\setminus\{x,y\}$1 (Pedrano et al., 20 Aug 2025). For graphs of order at least $S\setminus\{x,y\}$2, the paper proves the formula

$S\setminus\{x,y\}$3

(Pedrano et al., 20 Aug 2025).

The argument reported in the paper is direct. Choose one vertex $S\setminus\{x,y\}$4 and one vertex $S\setminus\{x,y\}$5, and set $S\setminus\{x,y\}$6. Because every vertex of $S\setminus\{x,y\}$7 is adjacent to every vertex of $S\setminus\{x,y\}$8 in the join, this 2-element set dominates the whole graph (Pedrano et al., 20 Aug 2025). To verify the 2-movability condition, the proof selects another pair $S\setminus\{x,y\}$9, $u,v\in V(G)\setminus S$0 and observes that replacing $u,v\in V(G)\setminus S$1 and $u,v\in V(G)\setminus S$2 by $u,v\in V(G)\setminus S$3 and $u,v\in V(G)\setminus S$4 again yields a dominating set (Pedrano et al., 20 Aug 2025). Since $u,v\in V(G)\setminus S$5 by the general lower bound, equality follows (Pedrano et al., 20 Aug 2025).

The result is especially clean because it is independent of the internal structure of $u,v\in V(G)\setminus S$6 and $u,v\in V(G)\setminus S$7, provided both sides are nontrivial. In the summary of contributions, this is described as a complete exact formula for the join (Pedrano et al., 20 Aug 2025).

A related but stricter total-domination analogue is established for joins as well: if $u,v\in V(G)\setminus S$8 and $u,v\in V(G)\setminus S$9 are graphs of order at least $x$0, then

$x$1

(Pedrano et al., 14 Aug 2025). The parallel formulas suggest that the join operation collapses both movable-domination parameters to the minimum nontrivial value under the stated order conditions.

4. Exact value for the corona of graphs

The second principal computation concerns the corona $x$2, where for each vertex $x$3 one attaches a copy $x$4 of $x$5 and joins $x$6 to every vertex of that copy (Pedrano et al., 20 Aug 2025). If $x$7 has $x$8 vertices, the construction therefore contains $x$9 attached copies of $y$0 (Pedrano et al., 20 Aug 2025).

For connected graphs satisfying

$y$1

the paper proves the exact formula

$y$2

(Pedrano et al., 20 Aug 2025). The proof proceeds in two directions.

For the upper bound, for each $y$3 one chooses a minimum dominating set $y$4 of the copy $y$5, and then sets

$y$6

The set $y$7 dominates each attached copy $y$8, and hence dominates the entire corona (Pedrano et al., 20 Aug 2025). To verify 2-movability, the proof considers separately the case where two removed vertices lie in the same copy and the case where they lie in different copies, and in both cases suitable replacements preserve domination (Pedrano et al., 20 Aug 2025). This yields

$y$9

(Pedrano et al., 20 Aug 2025).

For the lower bound, the proof assumes a smaller 2-movable dominating set $(S\setminus\{x,y\})\cup\{u,v\}$0 exists and divides into the cases $(S\setminus\{x,y\})\cup\{u,v\}$1 and $(S\setminus\{x,y\})\cup\{u,v\}$2 (Pedrano et al., 20 Aug 2025). Using internal restrictions on the intersection of $(S\setminus\{x,y\})\cup\{u,v\}$3 with an attached copy $(S\setminus\{x,y\})\cup\{u,v\}$4, the argument shows that some copy would then contain a dominating set smaller than $(S\setminus\{x,y\})\cup\{u,v\}$5, contradicting the definition of $(S\setminus\{x,y\})\cup\{u,v\}$6 (Pedrano et al., 20 Aug 2025). Equality follows.

The paper also derives the special case

$(S\setminus\{x,y\})\cup\{u,v\}$7

for connected $(S\setminus\{x,y\})\cup\{u,v\}$8 of order at least $(S\setminus\{x,y\})\cup\{u,v\}$9, using the identity $\gamma_m^2(G)$0 (Pedrano et al., 20 Aug 2025).

5. Internal lemmas for corona structure

The corona formula is supported by two structural lemmas that constrain how a 2-movable dominating set intersects each attached copy of $\gamma_m^2(G)$1 (Pedrano et al., 20 Aug 2025). These lemmas show that domination in the corona cannot be treated purely globally; it must be certified copy by copy.

The first lemma states that if $\gamma_m^2(G)$2 is a dominating set of $\gamma_m^2(G)$3 and $\gamma_m^2(G)$4, then

$\gamma_m^2(G)$5

is a dominating set of $\gamma_m^2(G)$6 provided $\gamma_m^2(G)$7 (Pedrano et al., 20 Aug 2025). This is used to force domination within attached copies whenever the corresponding base vertex is absent from the dominating set.

The second lemma states that if $\gamma_m^2(G)$8 is a 2-movable dominating set of $\gamma_m^2(G)$9, then for each attached copy $S\subseteq V(G)$00, the restriction $S\subseteq V(G)$01 inherits a corresponding movability property inside that copy (Pedrano et al., 20 Aug 2025). More specifically, for $S\subseteq V(G)$02, one of three alternatives holds:

1. $S\subseteq V(G)$03 is a dominating set of $S\subseteq V(G)$04, or
2. there exist replacement vertices $S\subseteq V(G)$05 outside $S\subseteq V(G)$06 such that

$S\subseteq V(G)$07

dominates $S\subseteq V(G)$08, or

1. there exists a vertex $S\subseteq V(G)$09 adjacent to $S\subseteq V(G)$10 such that

$S\subseteq V(G)$11

dominates $S\subseteq V(G)$12 (Pedrano et al., 20 Aug 2025).

These lemmas are technical, but they are the mechanism by which the global parameter $S\subseteq V(G)$13 is reduced to the ordinary domination number $S\subseteq V(G)$14 on each copy. A plausible implication is that corona graphs separate the “repair” phenomenon into independent local modules indexed by $S\subseteq V(G)$15.

6. Related frameworks, interpretation, and nearby variants

The most closely related broader framework is reconfiguration of dominating sets. The paper “Reconfiguration of Dominating Sets” studies the $S\subseteq V(G)$16-dominating graph $S\subseteq V(G)$17, whose vertices are dominating sets of size at most $S\subseteq V(G)$18 and whose edges connect dominating sets differing by exactly one vertex (Suzuki et al., 2014). That paper does not explicitly define or analyze a 2-movable dominating set, but it provides the nearest general framework for understanding domination under local moves (Suzuki et al., 2014).

Within that framework, a vertex $S\subseteq V(G)$19 is deletable if $S\subseteq V(G)$20 is still a dominating set, and the paper states that $S\subseteq V(G)$21 is deletable if and only if $S\subseteq V(G)$22 has at least one neighbour in $S\subseteq V(G)$23 and no private neighbour (Suzuki et al., 2014). This notion is not the same as 2-movability, but it is directly related to the first branch of the 2-movable condition, where removal preserves domination without repair.

The reconfiguration results also show that local flexibility can be limited even when some slack in set size is allowed. The same paper proves that $S\subseteq V(G)$24 is not necessarily connected and constructs graphs for which $S\subseteq V(G)$25 (Suzuki et al., 2014). This suggests that the existence of local repair conditions for a single dominating set should not be conflated with global connectivity or short paths in the full reconfiguration graph.

Another nearby but distinct framework is that of DD$S\subseteq V(G)$26-graphs, involving a disjoint dominating set $S\subseteq V(G)$27 and a disjoint 2-dominating set $S\subseteq V(G)$28 (Rana et al., 2023). That work studies domination, 2-domination, and their disjoint coexistence, not movable domination directly (Rana et al., 2023). Its relevance is conceptual rather than definitional: both settings concern strengthened domination requirements, but DD$S\subseteq V(G)$29 imposes disjointness and a 2-domination condition rather than pairwise deletability or replacement.

Finally, the literature already contains a stricter total-domination extension. In 2-movable total domination, a non-empty $S\subseteq V(G)$30 must be a total dominating set and satisfy the analogous two-vertex removal-or-replacement condition, with minimum size denoted $S\subseteq V(G)$31 (Pedrano et al., 14 Aug 2025). The comparison

$S\subseteq V(G)$32

makes clear that 2-movable domination sits between ordinary domination and its total-domination analogue in strength (Pedrano et al., 14 Aug 2025).

7. Examples and significance of the parameter

The paper includes an example graph distinguishing ordinary domination from 2-movable domination (Pedrano et al., 20 Aug 2025). In that example, $S\subseteq V(G)$33 is identified as a $S\subseteq V(G)$34-set, while $S\subseteq V(G)$35 is a $S\subseteq V(G)$36-set and $S\subseteq V(G)$37 is another $S\subseteq V(G)$38-set (Pedrano et al., 20 Aug 2025). The stated purpose of the example is to show that a 2-movable dominating set can be strictly larger than a minimum dominating set (Pedrano et al., 20 Aug 2025).

The currently available exact formulas can be summarized as follows.

Graph class	Condition	Value
Join $S\subseteq V(G)$39	$S\subseteq V(G)$40	$S\subseteq V(G)$41
Corona $S\subseteq V(G)$42	$S\subseteq V(G)$43 connected, $S\subseteq V(G)$44	$S\subseteq V(G)$45
Special case $S\subseteq V(G)$46	$S\subseteq V(G)$47 connected, order at least $S\subseteq V(G)$48	$S\subseteq V(G)$49

Taken together, these results show two sharply different behaviors (Pedrano et al., 20 Aug 2025). The join forces the parameter down to $S\subseteq V(G)$50 under nontrivial two-sided conditions, whereas the corona scales it by the number of attached copies and the ordinary domination number of the attached graph. This suggests that 2-movable domination is highly sensitive to whether graph operations create universal cross-adjacency, as in joins, or localized attachment modules, as in coronas.

At present, the explicit theory in the cited literature is concentrated on these operations and on conceptual comparisons with related notions. The available results therefore define the parameter clearly and establish its first exact formulas, while leaving broader structural, algorithmic, and extremal questions open in the current record (Pedrano et al., 20 Aug 2025).