Minimal Perfect Roman Domination

• Minimal Perfect Roman Dominating Functions are a graph theoretical concept defined by assigning only values 0 and 2 to vertices, ensuring a dominating set remains invariant under vertex deletion.
• They enforce a unique structure where every vertex in the dominating set has at least three private neighbors, maintaining stability and minimality across the graph.
• Structural results in trees and extremal bounds provide efficient algorithmic constructions and tight theoretical limits useful for advanced graph theory applications.

A minimal perfect Roman dominating function is a critical concept in the paper of domination in graphs, merging the combinatorial structure of minimality and uniqueness with the robustness properties characteristic of Roman domination. This article provides a comprehensive survey of the precise definitions, structural theorems, extremal results, algorithmic implications, and open directions associated with minimal perfect Roman dominating functions, focusing on the class $\mathcal{R}_{UVR}$ as systematically developed in contemporary research.

1. Definitions and Fundamental Concepts

Let $G=(V,E)$ be a finite simple graph. A Roman dominating function (RDF) is a function

$f: V \longrightarrow \{0,1,2\}$

such that every vertex $v$ with $f(v) = 0$ has at least one neighbor $u$ with $f(u) = 2$. The weight of an RDF $f$ is

$w(f) = \sum_{v \in V} f(v) = |V_1^f| + 2|V_2^f|,$

where $V_i^f = \{ v \in V : f(v) = i \}$ for $i \in \{0,1,2\}$. The Roman domination number of $G$, denoted $\gamma_R(G)$, is the minimum weight of an RDF on $G$: $$\gamma_R(G) = \min \{ w(f) : f \text{ is an RDF on } G \}.$$ An RDF of weight $\gamma_R(G)$ is called a $\gamma_R$–function.

A graph $G$ belongs to the class $\mathcal{R}_{UVR}$ if the Roman domination number is invariant under the deletion of any vertex: $$\forall v \in V(G): \quad \gamma_R(G - v) = \gamma_R(G).$$ Such graphs exhibit a form of "minimal perfection" in Roman domination, as their $\gamma_R$–functions possess strong global structural uniqueness and resistance to vertex removal (Samodivkin, 2015).

2. Structural Characterization of the Class $\mathcal{R}_{UVR}$

2.1 Uniqueness and Form of Minimal Roman Functions

Every $G \in \mathcal{R}_{UVR}$ admits a $\gamma_R$–function $f$ with $V_1^f = \emptyset$; that is, only the values $0$ and $2$ appear. In this case, $V_2^f$ must be a dominating set (a $\gamma$–set), and further, every $v \in V_2^f$ must have at least three private neighbors in $V_0^f$: $$\forall v \in V_2^f,\quad |\operatorname{pn}_G[v, V_2^f]| \ge 3.$$ Conversely, if a graph $G$ has a $\gamma_R$–function with $V_1 = \emptyset$ and every $v \in V_2$ has at least three private neighbors, then $G \in \mathcal{R}_{UVR}$ (Samodivkin, 2015).

2.2 Trees in $\mathcal{R}_{UVR}$: Explicit Construction

The subclass of trees in $\mathcal{R}_{UVR}$ is characterized in full:

Let $T$ be a tree with $|V(T)| \ge 3$. The following are equivalent:

• $T \in \mathcal{R}_{UVR}$.
• $T$ has exactly one $\gamma_R$–function $f$, with $V_1^f = \varnothing$, $V_2^f$ independent, and each $v \in V_2^f$ has exactly three private neighbors.
• $T$ has a unique $\gamma$–set $D$ (dominating set of minimum size), with $D$ independent, and each $v \in D$ has three private neighbors.
• $T$ can be built from a labeled $K_{1,2}$ (center labeled $B$, leaves labeled $A$) by iterated application of four operations (O1–O4), each precisely maintaining the structural property above.

The operations (O1–O4) use elementary attachments of paths and stars and preserve the labeling invariant: the set of $B$-labeled vertices is always the unique $\gamma$–set, and this labeling directly prescribes the minimal perfect Roman dominating function (Samodivkin, 2015).

3. Extremal Bounds and Packing Properties

If $G \in \mathcal{R}_{UVR}$ is a connected graph with $n$ vertices, the Roman domination number satisfies

$\gamma_R(G) \le \frac{2}{3} n.$

Equality holds if and only if every $\gamma_R$–function $f$ has $V_2^f$ as an efficient dominating set (packing) of degree 2 vertices. In trees, only those constructed using repeated O1 operations (with no $C$-labels) attain the extremal ratio, and the underlying set $V_2^f$ is a maximal packing of degree-2 vertices (Samodivkin, 2015).

Further, in any $G \in \mathcal{R}_{UVR}$, every minimum Roman function yields $V_2^f$ as an efficient packing, and each $v \in V_2^f$ has at least three private neighbors.

The Roman bondage number $b_R(G)$, the minimum number of edges whose deletion increases $\gamma_R(G)$, satisfies for all $G \in \mathcal{R}_{UVR}$

$b_R(G) \le \delta(G)$

where $\delta(G)$ is the minimum degree of $G$. This bound is tight, as removing all edges incident to a vertex of minimum degree guarantees an increase in $\gamma_R$ (Samodivkin, 2015).

4. Relationships with Criticality, Uniqueness, and Minimality

The notion of minimal perfect Roman domination in $\mathcal{R}_{UVR}$ is directly related to the unchanging Roman domination class under vertex deletion, distinguishing it within the broader landscape of criticality in domination invariants (Samodivkin, 2017). While $k$-criticality in other settings refers to decreasing the domination number under removal of any $k$ vertices, $\mathcal{R}_{UVR}$ captures the opposite: absolute invariance under all single-vertex deletions.

A minimal perfect Roman dominating function on $G \in \mathcal{R}_{UVR}$ is unique, and this minimality is tied to the property that no vertex can simultaneously serve as a $1$- or $2$-label, except precisely as dictated by the rigid $\gamma$–set construction (Samodivkin, 2015). In contrast, in the class $\mathcal{R}_{CVR}$ (the changing class), removal of any vertex strictly decreases $\gamma_R$.

Minimal perfect Roman dominating functions thus form the extreme point of uniqueness and stability in the Roman domination spectrum, directly associating them with highly symmetric and rigid graph structures.

5. Algorithmic and Combinatorial Consequences

The explicit labeling scheme for trees—along with the structural characterization—enables algorithmic construction of all minimal perfect Roman dominating functions in $\mathcal{R}_{UVR}$-trees and yields fast (linear time) recognition algorithms for such trees.

For any $\mathcal{R}_{UVR}$-tree, the Roman bondage number is exactly one, since the removal of any edge connecting two $0$-labeled vertices under the unique $\gamma_R$–function produces another $\mathcal{R}_{UVR}$-tree (Samodivkin, 2015).

When considering the behavior under products and Mycieleskian constructions, related studies further clarify that $\mathcal{R}_{UVR}$-type graphs correspond to those "special Roman" graphs in which the structure of the minimal dominating function is preserved under certain graph operations (Kazemi, 2011). The efficient packing property of $V_2^f$ is also critical in bounding and calculating Roman domination parameters under products (Martinez et al., 2021).

6. Illustrative Examples and Extremal Cases

A canonical example in $\mathcal{R}_{UVR}$ is $C_6$, the 6-cycle. Its minimal perfect Roman dominating function assigns the value $2$ to two antipodal vertices (and $0$ to the rest), and this minimal function persists with the same weight after any single vertex removal. In contrast, $P_3$ (the path of three vertices) does not belong to $\mathcal{R}_{UVR}$, as its Roman domination number decreases under vertex deletion (Samodivkin, 2015).

Among trees, those constructed with repeated O1 operations, i.e., chains of $K_{1,2}$ attachments, yield extremal $2/3$ ratios, and their unique minimal perfect Roman dominating function is explicitly determined by the construction.

7. Open Problems and Research Directions

Current open questions regarding minimal perfect Roman dominating functions include:

• Complete classification of unicyclic graphs in $\mathcal{R}_{UVR}$.
• Determination of the maximal size of an $n$-vertex $G \in \mathcal{R}_{UVR}$ with a given Roman domination number $\gamma_R(G) = k$.
• Refinement of the upper $\frac{2}{3}n$ bound in the presence of additional degree constraints.

A plausible implication is that further exploration of the interplay between efficient packing, degree constraints, and domination invariants in minimal perfect Roman domination may yield new extremal families outside of the current constructive classes.

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References

• "Roman domination in graphs: the class $\mathcal{R}_{UVR}$" (Samodivkin, 2015)
• "Roman domination: changing, unchanging, $\gamma_R$-graphs" (Samodivkin, 2017)
• "Roman domination and Mycieleki's structure in graphs" (Kazemi, 2011)
• "Roman domination in direct product graphs and rooted product graphs" (Martinez et al., 2021)