2-Movable Domination in Graphs

• 2-movable domination is a graph parameter defined on connected graphs, where a dominating set must allow a valid pairwise replacement to maintain its dominating property.
• For join operations, the 2-movable domination number collapses to 2 by choosing one vertex from each factor, while for coronas it scales as |V(G)| multiplied by the domination number of the attached graph.
• These findings highlight distinct structural consequences of graph operations on domination properties and open avenues for further algorithmic and theoretical exploration.

Searching arXiv for the specified paper and related context. The 2-movable domination number is a graph parameter defined for a connected, finite, simple graph $G$ by imposing a pairwise replacement condition on dominating sets. A non-empty set $S \subseteq V(G)$ is a 2-movable dominating set if $S$ is dominating and, for every pair $x,y \in S$, either $S \setminus \{x,y\}$ remains dominating, 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 dominating. The minimum cardinality of such a set is the 2-movable domination number, denoted $\gamma_m^2(G)$, and a minimum set is a $S \subseteq V(G)$0-set. The parameter is studied in detail for graph joins and coronas in “On 2-Movable Domination in the Join and Corona of Graphs” (Pedrano et al., 20 Aug 2025).

1. Definition and formal framework

Let $S \subseteq V(G)$1 be a connected graph with vertex set $S \subseteq V(G)$2. A set $S \subseteq V(G)$3 is a dominating set if every vertex in $S \subseteq V(G)$4 has a neighbor in $S \subseteq V(G)$5; equivalently, the closed neighborhood of $S \subseteq V(G)$6 covers $S \subseteq V(G)$7. The classical domination number $S \subseteq V(G)$8 is the minimum cardinality of a dominating set.

Definition 2.1 introduces 2-movable domination as a strengthening of domination. A non-empty set $S \subseteq V(G)$9 is a 2-movable dominating set if it is dominating and for every pair $S$0, one of the following holds:

1. $S$1 is a dominating set in $S$2; or
2. there exist $S$3 such that $S$4 is adjacent to $S$5, $S$6 is adjacent to $S$7, and $S$8 is a dominating set.

The minimum size of such a set is $S$9, and any 2-movable dominating set of size $x,y \in S$0 is a $x,y \in S$1-set (Pedrano et al., 20 Aug 2025).

This definition encodes a robustness condition under simultaneous loss or relocation of two vertices from the dominating set. Unlike ordinary domination, which is static, 2-movable domination requires that every ordered choice of two members of the set can be removed while preserving domination directly or after admissible local substitutions. A plausible implication is that the parameter is naturally suited to settings in which redundancy under paired failures or paired relocations is structurally relevant, although the paper itself develops the theory in purely graph-theoretic terms.

2. Basic inequalities and conceptual position

Two universal constraints are established for connected graphs of order at least $x,y \in S$2. First, Remark 3.1 states that

$x,y \in S$3

Second, Theorem 3.2 proves that

$x,y \in S$4

The second inequality is immediate from the fact that every 2-movable dominating set is, in particular, a dominating set (Pedrano et al., 20 Aug 2025).

These bounds place 2-movable domination strictly above classical domination in the sense of feasibility requirements. The lower bound $x,y \in S$5 excludes trivial one-vertex solutions on connected graphs of order at least $x,y \in S$6, even when $x,y \in S$7.

The paper also recalls 1-movable domination from Blair, Gera, and Horton. However, the explicit inequality proved in the paper is $x,y \in S$8. The statement that $x,y \in S$9 is described as natural and consistent with that literature, but it is not given as a proved theorem in this work. This distinction is important: 2-movable domination should not be conflated either with classical domination or with previously studied 1-movable domination.

A standard example illustrating strict separation from $S \setminus \{x,y\}$0 is the star $S \setminus \{x,y\}$1 for $S \setminus \{x,y\}$2. Here $S \setminus \{x,y\}$3, but $S \setminus \{x,y\}$4. A minimal 2-movable dominating set must avoid the center, because if the center were included, then for the pair consisting of the center and a leaf, the required replacement adjacent to the leaf could not be chosen outside the set. By contrast, the set of all $S \setminus \{x,y\}$5 leaves is dominating and 2-movable: for any two leaves $S \setminus \{x,y\}$6, choosing $S \setminus \{x,y\}$7 center yields $S \setminus \{x,y\}$8, which dominates the star. No proper subset of leaves dominates all leaves, since each leaf is adjacent only to the center.

3. Behavior under join and corona operations

The paper studies two graph operations in detail.

The join of disjoint graphs $S \setminus \{x,y\}$9 and $u,v \in V(G)\setminus S$0, denoted $u,v \in V(G)\setminus S$1 or $u,v \in V(G)\setminus S$2, is defined by

$u,v \in V(G)\setminus S$3

$u,v \in V(G)\setminus S$4

Thus every vertex of $u,v \in V(G)\setminus S$5 is adjacent to every vertex of $u,v \in V(G)\setminus S$6.

The corona of graphs, denoted $u,v \in V(G)\setminus S$7 and written $u,v \in V(G)\setminus S$8 in the paper, is formed by taking, for each $u,v \in V(G)\setminus S$9, a distinct copy $x$0 of $x$1 and joining $x$2 to all vertices of $x$3. Formally,

$x$4

$x$5

For a vertex $x$6, the induced subgraph on $x$7 is denoted $x$8 (Pedrano et al., 20 Aug 2025).

These operations have sharply different effects on $x$9. The join creates universal cross-adjacency between the two factors, which collapses the 2-movable domination number to its minimum possible value under the standing lower bound. The corona instead localizes domination requirements into the attached copies $y$0, producing a multiplicative formula in terms of $y$1 and $y$2.

The contrast is structurally informative. In joins, a pair of vertices chosen from opposite sides can dominate globally and can be replaced globally because of complete cross-edges. In coronas, domination is fiberwise: each copy $y$3 must be controlled locally, and pairwise movements either remain within a fiber or are mediated by the apex vertex $y$4.

4. Exact results for joins

Theorem 3.3 gives a complete answer for joins: if $y$5 and $y$6 are graphs of order at least $y$7, then

$y$8

Since both graphs have order at least $y$9, the join has order at least $(S \setminus \{x,y\}) \cup \{u,v\}$0, so the universal lower bound applies. Equality is obtained by choosing one vertex $(S \setminus \{x,y\}) \cup \{u,v\}$1 and one vertex $(S \setminus \{x,y\}) \cup \{u,v\}$2 and taking $(S \setminus \{x,y\}) \cup \{u,v\}$3 (Pedrano et al., 20 Aug 2025).

The mechanism is direct. Because every vertex of $(S \setminus \{x,y\}) \cup \{u,v\}$4 is adjacent to every vertex of $(S \setminus \{x,y\}) \cup \{u,v\}$5, the vertex $(S \setminus \{x,y\}) \cup \{u,v\}$6 dominates all of $(S \setminus \{x,y\}) \cup \{u,v\}$7 and the vertex $(S \setminus \{x,y\}) \cup \{u,v\}$8 dominates all of $(S \setminus \{x,y\}) \cup \{u,v\}$9, so $\gamma_m^2(G)$0 is dominating. For the only relevant pair $\gamma_m^2(G)$1, choose $\gamma_m^2(G)$2 and $\gamma_m^2(G)$3. Then

$\gamma_m^2(G)$4

which is again dominating by the same join property, and the adjacency requirements hold across the join.

The theorem immediately yields exact values for standard joined families:

Graph family	Condition	Value of $\gamma_m^2(G)$5
$\gamma_m^2(G)$6	$\gamma_m^2(G)$7	$\gamma_m^2(G)$8
$\gamma_m^2(G)$9	$S \subseteq V(G)$00	$S \subseteq V(G)$01
$S \subseteq V(G)$02	$S \subseteq V(G)$03	$S \subseteq V(G)$04

The paper also notes the identification

$S \subseteq V(G)$05

and from Corollary 3.1 concludes that

$S \subseteq V(G)$06

For example, since $S \subseteq V(G)$07, one has $S \subseteq V(G)$08.

A related example, presented as an example argument rather than a stated theorem, is the complete graph $S \subseteq V(G)$09 with $S \subseteq V(G)$10. Taking any two distinct vertices $S \subseteq V(G)$11 gives a dominating set, and for the pair $S \subseteq V(G)$12 one may choose two further vertices $S \subseteq V(G)$13 so that $S \subseteq V(G)$14 still dominates $S \subseteq V(G)$15. Hence $S \subseteq V(G)$16, in agreement with the general lower bound. Likewise, for complete bipartite graphs $S \subseteq V(G)$17 with $S \subseteq V(G)$18, choosing one vertex from each bipartition yields $S \subseteq V(G)$19 by the same replacement pattern across the bipartition.

5. Exact results for coronas

The corona case is governed by two preparatory lemmas and one exact formula. Lemma 3.4 states that if $S \subseteq V(G)$20 is a dominating set of $S \subseteq V(G)$21 and $S \subseteq V(G)$22 with $S \subseteq V(G)$23, then

$S \subseteq V(G)$24

is a dominating set of $S \subseteq V(G)$25.

Lemma 3.5 describes the local 2-movable behavior inside a fiber. If $S \subseteq V(G)$26 is a 2-movable dominating set of $S \subseteq V(G)$27, $S \subseteq V(G)$28, $S \subseteq V(G)$29, and $S \subseteq V(G)$30, then one of the following holds:

1. $S \subseteq V(G)$31 is a dominating set of $S \subseteq V(G)$32; or
2. there exist $S \subseteq V(G)$33 such that $S \subseteq V(G)$34 and $S \subseteq V(G)$35 is a dominating set of $S \subseteq V(G)$36; or
3. there exists $S \subseteq V(G)$37 such that $S \subseteq V(G)$38 and $S \subseteq V(G)$39 is a dominating set of $S \subseteq V(G)$40.

These lemmas formalize the fact that, in the corona, the 2-movable condition projects to the induced subgraphs $S \subseteq V(G)$41.

Theorem 3.6 then proves the exact formula: if $S \subseteq V(G)$42 and $S \subseteq V(G)$43 are connected graphs with $S \subseteq V(G)$44, then

$S \subseteq V(G)$45

(Pedrano et al., 20 Aug 2025).

The order condition may be written as

$S \subseteq V(G)$46

In particular, it holds if $S \subseteq V(G)$47, or if $S \subseteq V(G)$48 with $S \subseteq V(G)$49.

The upper bound is constructive. For each $S \subseteq V(G)$50, let $S \subseteq V(G)$51 be a $S \subseteq V(G)$52-set in the copy $S \subseteq V(G)$53, and define

$S \subseteq V(G)$54

This set dominates every fiber $S \subseteq V(G)$55, hence the whole corona. For the 2-movable property, two cases are distinguished. If the chosen pair lies in the same fiber $S \subseteq V(G)$56, the replacements are performed inside $S \subseteq V(G)$57 using local adjacency in the fiber and, where needed, the apex $S \subseteq V(G)$58. If the chosen pair lies in different fibers $S \subseteq V(G)$59 and $S \subseteq V(G)$60, replacing them by $S \subseteq V(G)$61 and $S \subseteq V(G)$62 preserves domination in those fibers because each apex is adjacent to all vertices of its own copy.

The lower bound excludes any 2-movable dominating set $S \subseteq V(G)$63 with cardinality less than $S \subseteq V(G)$64. If $S \subseteq V(G)$65, some fiber $S \subseteq V(G)$66 would be completely uncovered, contradicting domination. If $S \subseteq V(G)$67, then some fiber satisfies $S \subseteq V(G)$68, contradicting Lemma 3.4 or Lemma 3.5 after the required local 2-movable replacements are examined.

Several immediate consequences are listed:

Corona family	Exact value
$S \subseteq V(G)$69	$S \subseteq V(G)$70
$S \subseteq V(G)$71	$S \subseteq V(G)$72
$S \subseteq V(G)$73	$S \subseteq V(G)$74

The same formula also holds for trees $S \subseteq V(G)$75, namely $S \subseteq V(G)$76.

Concrete examples include:

• if $S \subseteq V(G)$77 and $S \subseteq V(G)$78, then for any connected $S \subseteq V(G)$79 with $S \subseteq V(G)$80,

$S \subseteq V(G)$81

• if $S \subseteq V(G)$82 and $S \subseteq V(G)$83, then

$S \subseteq V(G)$84

6. Constructions, examples, and computational implications

The proofs for joins and coronas are constructive and provide explicit $S \subseteq V(G)$85-sets.

For joins, the construction is minimal and uniform: choose one vertex from each factor. Any set

$S \subseteq V(G)$86

is a $S \subseteq V(G)$87-set when both factors have order at least $S \subseteq V(G)$88. The admissible replacements for the pair $S \subseteq V(G)$89 are any

$S \subseteq V(G)$90

and $S \subseteq V(G)$91 remains dominating because of the complete cross-adjacency.

For coronas, the construction is fiberwise: in each copy $S \subseteq V(G)$92, choose a $S \subseteq V(G)$93-set $S \subseteq V(G)$94 and let

$S \subseteq V(G)$95

If a pair lies in the same fiber, the replacement is handled locally inside $S \subseteq V(G)$96 as encoded by Lemma 3.5. If a pair lies in different fibers, each selected vertex can be replaced by the corresponding apex. This yields a systematic construction of $S \subseteq V(G)$97-sets for every corona covered by Theorem 3.6 (Pedrano et al., 20 Aug 2025).

Small illustrative examples make the mechanism explicit.

In the join example $S \subseteq V(G)$98 with vertices $S \subseteq V(G)$99 and $S$00 with vertices $S$01, the graph $S$02 is $S$03. The set $S$04 is a $S$05-set, and for the pair $S$06 one may replace by $S$07 and $S$08, since $S$09 dominates $S$10.

In the corona example with $S$11 having vertices $S$12 and $S$13 having vertices $S$14, one has $S$15. Choosing one dominating vertex in each copy gives a set of size $S$16, which is a $S$17-set by Theorem 3.6. If the moved pair comes from different fibers, the replacements are the corresponding apex vertices $S$18 and $S$19.

In the star $S$20 with center $S$21 and leaves $S$22, the set of all leaves is a $S$23-set. For any pair $S$24, taking $S$25 gives

$S$26

which dominates the whole graph. This example emphasizes that 2-movable domination can be much larger than ordinary domination.

The paper does not provide algorithmic complexity analyses or general algorithms for computing $S$27 on arbitrary graphs. It does state, however, that the constructive proofs of Theorems 3.3 and 3.6 yield immediate linear-time constructions of $S$28-sets for joins and coronas when $S$29-sets are known.

7. Scope, limitations, and natural directions

The results establish two contrasting structural laws. For joins, 2-movable domination collapses to the minimum possible value allowed by the universal bound:

$S$30

whenever both factors have order at least $S$31. For coronas, the parameter scales multiplicatively:

$S$32

These formulas show that the effect of graph operations on $S$33 is highly operation-sensitive (Pedrano et al., 20 Aug 2025).

The work also delineates what is not established. The paper does not provide values of $S$34 for paths $S$35 or cycles $S$36 themselves. It only gives the general lower relation

$S$37

together with the classical values

$S$38

Accordingly, one should not infer exact 2-movable domination numbers for these base families from the corona formulas.

No explicit open problems or conjectures are stated, but several natural directions are identified in the surrounding discussion. These include determining $S$39 for broader graph classes such as paths, cycles, general trees, and bipartite graphs beyond the join and corona settings; obtaining tight bounds in terms of degree parameters; and studying the algorithmic complexity of recognizing and computing $S$40. This suggests that the paper is foundational with respect to graph operations rather than exhaustive with respect to general graph classes.