---
title: 2-Movable Dominating Set in Graphs
url: https://www.emergentmind.com/topics/2-movable-dominating-set
type: topic
---

# 2-Movable Dominating Set in Graphs

Searching arXiv for recent 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 [2509.05301]. The associated minimum cardinality is the **2-movable domination number**, denoted \(\gamma_m^2(G)\), and a minimum such set is a \(\gamma_m^2\)-set [2509.05301]. 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 [2509.05301].

## 1. Formal definition and parameterization

The underlying setting is that of connected, finite, simple, undirected graphs [2509.05301]. Ordinary domination requires that every vertex outside a set \(S\) be adjacent to at least one vertex of \(S\). The 2-movable condition adds a uniform two-vertex resilience requirement: after deleting any two vertices of \(S\), domination must either persist immediately or be restorable by inserting two outside vertices, one adjacent to each removed vertex [2509.05301].

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

1. \(S\) is a dominating set, and  
2. for every pair \(x,y \in S\), either  
   \[
   S\setminus\{x,y\}
   \]
   is a dominating set in \(G\), or there exist \(u,v \in V(G)\setminus S\) such that \(u\) is adjacent to \(x\), \(v\) is adjacent to \(y\), and  
   \[
   (S\setminus\{x,y\})\cup\{u,v\}
   \]
   is a dominating set in \(G\) [2509.05301].

The minimum cardinality of a 2-movable dominating set is the **2-movable domination number** \(\gamma_m^2(G)\), and a minimum 2-movable dominating set is called a **\(\gamma_m^2\)-set** [2509.05301]. 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 [2509.05301].

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 [2509.05301]. The paper recalls ordinary domination and 1-movable domination, and notes that 2-movable domination is more restrictive than ordinary domination [2509.05301].

For any connected graph \(G\) of order \(n\ge 4\), the paper states the basic lower bound
\[
\gamma_m^2(G)\ge 2
\]
[2509.05301]. It also proves that 2-movable domination is at least as restrictive as ordinary domination and 1-movable domination, summarized in the hierarchy
\[
\gamma(G)\le \gamma_m^1(G)\le \gamma_m^2(G)
\]
in the intended interpretation reported in the paper summary [2509.05301].

The article on **2-movable total domination** gives an analogous comparison in the total-domination setting: if \(S\) is a 2-movable total dominating set of \(G\), then \(S\) is also a 2-movable dominating set, so
\[
\gamma_m^2(G)\le \gamma_{mt}^2(G)
\]
[2508.10952]. 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 [2509.05301].

## 3. Exact value for the join of graphs

One of the main exact results concerns the **join** \(G+H\), formed from disjoint copies of \(G\) and \(H\) by adding all edges between every vertex of \(G\) and every vertex of \(H\) [2509.05301]. For graphs of order at least \(2\), the paper proves the formula
\[
\gamma_m^2(G+H)=2
\]
[2509.05301].

The argument reported in the paper is direct. Choose one vertex \(u\in V(G)\) and one vertex \(v\in V(H)\), and set \(S=\{u,v\}\). Because every vertex of \(G\) is adjacent to every vertex of \(H\) in the join, this 2-element set dominates the whole graph [2509.05301]. To verify the 2-movability condition, the proof selects another pair \(u'\in V(G)\), \(v'\in V(H)\) and observes that replacing \(u\) and \(v\) by \(u'\) and \(v'\) again yields a dominating set [2509.05301]. Since \(\gamma_m^2(G+H)\ge 2\) by the general lower bound, equality follows [2509.05301].

The result is especially clean because it is independent of the internal structure of \(G\) and \(H\), provided both sides are nontrivial. In the summary of contributions, this is described as a complete exact formula for the join [2509.05301].

A related but stricter total-domination analogue is established for joins as well: if \(G\) and \(H\) are graphs of order at least \(2\), then
\[
\gamma_{mt}^2(G+H)=2
\]
[2508.10952]. 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** \(G\cdot H\), where for each vertex \(x\in V(G)\) one attaches a copy \(H_x\) of \(H\) and joins \(x\) to every vertex of that copy [2509.05301]. If \(G\) has \(n\) vertices, the construction therefore contains \(n\) attached copies of \(H\) [2509.05301].

For connected graphs satisfying
\[
|V(G\cdot H)|\ge 4,
\]
the paper proves the exact formula
\[
\gamma_m^2(G\cdot H)=|V(G)|\,\gamma(H)
\]
[2509.05301]. The proof proceeds in two directions.

For the upper bound, for each \(x\in V(G)\) one chooses a minimum dominating set \(S_x\) of the copy \(H_x\), and then sets
\[
S=\bigcup_{x\in V(G)} S_x.
\]
The set \(S\) dominates each attached copy \(H_x\), and hence dominates the entire corona [2509.05301]. 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 [2509.05301]. This yields
\[
\gamma_m^2(G\cdot H)\le |V(G)|\,\gamma(H)
\]
[2509.05301].

For the lower bound, the proof assumes a smaller 2-movable dominating set \(T\) exists and divides into the cases \(\gamma(H)=1\) and \(\gamma(H)\ge 2\) [2509.05301]. Using internal restrictions on the intersection of \(T\) with an attached copy \(H_a\), the argument shows that some copy would then contain a dominating set smaller than \(\gamma(H)\), contradicting the definition of \(\gamma(H)\) [2509.05301]. Equality follows.

The paper also derives the special case
\[
\gamma_m^2(K_1+H)=\gamma(H)
\]
for connected \(H\) of order at least \(4\), using the identity \(K_1+H = K_1\cdot H\) [2509.05301].

## 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 \(H\) [2509.05301]. 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 \(T\) is a dominating set of \(G\cdot H\) and \(a\in V(G)\), then
\[
T_a = T\cap V(H_a)
\]
is a dominating set of \(H_a\) provided \(a\notin T\) [2509.05301]. 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 \(T\) is a 2-movable dominating set of \(G\cdot H\), then for each attached copy \(a+H_a\), the restriction \(T_a\) inherits a corresponding movability property inside that copy [2509.05301]. More specifically, for \(u,z\in T_a\), one of three alternatives holds:

1. \(T_a\setminus\{u,z\}\) is a dominating set of \(H_a\), or  
2. there exist replacement vertices \(x_a,x_u\) outside \(T_a\) such that  
   \[
   (T_a\setminus\{u,z\})\cup\{x_a,x_u\}
   \]
   dominates \(H_a\), or  
3. there exists a vertex \(x_u\notin T_a\) adjacent to \(u\) such that  
   \[
   (T_a\setminus\{u,z\})\cup\{x_u\}
   \]
   dominates \(H_a\) [2509.05301].

These lemmas are technical, but they are the mechanism by which the global parameter \(\gamma_m^2(G\cdot H)\) is reduced to the ordinary domination number \(\gamma(H)\) on each copy. A plausible implication is that corona graphs separate the “repair” phenomenon into independent local modules indexed by \(V(G)\).

## 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 \(k\)-dominating graph \(D_k(G)\), whose vertices are dominating sets of size at most \(k\) and whose edges connect dominating sets differing by exactly one vertex [1401.5714]. 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 [1401.5714].

Within that framework, a vertex \(v\in S\) is **deletable** if \(S\setminus\{v\}\) is still a dominating set, and the paper states that \(v\) is deletable if and only if \(v\) has at least one neighbour in \(S\) and no private neighbour [1401.5714]. 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 \(D_{\Gamma(G)+1}(G)\) is not necessarily connected and constructs graphs for which \(\operatorname{diam}(D_{\gamma(G)+1}(G))=\Omega(2^n)\) [1401.5714]. 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\(_2\)-graphs**, involving a disjoint dominating set \(D\) and a disjoint 2-dominating set \(D_2\) [2312.01149]. That work studies domination, 2-domination, and their disjoint coexistence, not movable domination directly [2312.01149]. Its relevance is conceptual rather than definitional: both settings concern strengthened domination requirements, but DD\(_2\) 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 \(T\subseteq V(G)\) must be a total dominating set and satisfy the analogous two-vertex removal-or-replacement condition, with minimum size denoted \(\gamma_{mt}^2(G)\) [2508.10952]. The comparison
\[
\gamma_m^2(G)\le \gamma_{mt}^2(G)
\]
makes clear that 2-movable domination sits between ordinary domination and its total-domination analogue in strength [2508.10952].

## 7. Examples and significance of the parameter

The paper includes an example graph distinguishing ordinary domination from 2-movable domination [2509.05301]. In that example, \(S_1=\{v_2,v_7,v_9\}\) is identified as a \(\gamma\)-set, while \(S_2=\{v_2,v_6,v_7,v_9\}\) is a \(\gamma_m^2\)-set and \(S_3=\{v_2,v_5,v_6,v_7,v_{12}\}\) is another \(\gamma_m^2\)-set [2509.05301]. The stated purpose of the example is to show that a 2-movable dominating set can be strictly larger than a minimum dominating set [2509.05301].

The currently available exact formulas can be summarized as follows.

| Graph class | Condition | Value |
|---|---|---|
| Join \(G+H\) | \(|V(G)|,|V(H)|\ge 2\) | \(\gamma_m^2(G+H)=2\) |
| Corona \(G\cdot H\) | \(G,H\) connected, \(|V(G\cdot H)|\ge 4\) | \(\gamma_m^2(G\cdot H)=|V(G)|\,\gamma(H)\) |
| Special case \(K_1+H\) | \(H\) connected, order at least \(4\) | \(\gamma_m^2(K_1+H)=\gamma(H)\) |

Taken together, these results show two sharply different behaviors [2509.05301]. The join forces the parameter down to \(2\) 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 [2509.05301].

Source: https://www.emergentmind.com/topics/2-movable-dominating-set