Roman Domination and γR-Excellent Graphs

• Roman domination is defined by labeling vertices with 0, 1, or 2 such that every 0-labeled vertex has a neighbor labeled 2, and the minimum labeling weight defines γR(G).
• γR-excellent graphs guarantee that every vertex can appear as a defender (labeled 1 or 2) in some minimum-weight Roman dominating function, highlighting robust network defense.
• Constructive characterizations using recursive tree operations reveal that UVR properties and status partitioning are central to maintaining stability under vertex or edge modifications.

A Roman dominating function (RDF) on a simple graph $G=(V,E)$ is a labeling $f:V\to\{0,1,2\}$ such that every vertex $v$ with $f(v)=0$ has a neighbor $u$ with $f(u)=2$. The weight of $f$ is $f(V)=\sum_{v\in V} f(v)$, and the Roman domination number $\gamma_R(G)$ is the minimum weight across all RDFs on $G$. An RDF achieving weight $\gamma_R(G)$ is called a $\gamma_R$-function. A graph $G$ is $\gamma_R$-excellent if for every vertex $x\in V$ there exists a $\gamma_R$-function $h_x$ with $h_x(x)\neq 0$; equivalently, every vertex is labeled $1$ or $2$ in some minimum-weight RDF. This property, and its structural manifestations in classes such as UVR (where for every vertex $v$, $\gamma(G-v)=\gamma(G)$ for domination number), have been thoroughly characterized for trees and related classes, revealing robust and "nice" behavior for network resilience and combinatorial structure (Samodivkin, 2016, Samodivkin, 2015, Samodivkin, 2017).

1. Roman Domination: Definitions and Fundamental Invariants

Let $G=(V,E)$ be a finite simple graph. The Roman domination number $\gamma_R(G)$ serves as a measure of minimal "reinforcement" required to defend all vertices, with labels representing defensive resources. The partition $V_i^f = \{v\in V \,:\, f(v) = i\}$ for $i=0,1,2$ provides structure for analysis. The classical domination number $\gamma(G)$ is defined as the cardinality of a smallest dominating set $D$ such that $N[D]=V$.

The Roman bondage number $b_R(G)$ is the minimum cardinality $|F|\subseteq E$ such that $\gamma_R(G-F) > \gamma_R(G)$; it quantifies the stability of $\gamma_R$ under edge deletions. The paper of these invariants on UVR graphs (those with $\gamma_R(G-v)=\gamma_R(G)$ for all $v\in V$) yields sharp bounds and tight characterizations: Specifically, for $G\in \mathcal{R}_{UVR}$ connected of order $n$, $\gamma_R(G)\leq (2/3)n$ with equality only for those graphs where all minimum RDFs are label-$1$-free, their label-$2$ sets form efficient dominating sets of degree 2 (Samodivkin, 2015).

2. $\gamma_R$-Excellent Graphs: Definition and Characterization

A graph $G$ is $\gamma_R$-excellent if every vertex $x\in V$ admits a minimum-weight RDF $h_x$ with $h_x(x)\neq 0$. Equivalently, no vertex is forced to be labeled 0 in all $\gamma_R$-functions. For trees, this excellence is characterized constructively via a quartet status labeling $S:V\to\{A,B,C,D\}$ and a set of recursive building operations ($O_1$ through $O_4$) starting from five "seed" trees $H_1,\ldots,H_5$ with specific labelings (Samodivkin, 2016):

• Status A: $V^{01}(T)$ (vertices that can be labeled 0 or 1 in some $\gamma_R$-function).
• Status D: $V^{012}(T)$ (vertices that can be labeled 0, 1, or 2).
• Status B: vertices in $V^{02}(T)$ of degree 2 with exactly one neighbor in $V^{02}(T)$.
• Status C: remaining $V^{02}(T)$-vertices.

A tree $T$ is $\gamma_R$-excellent if and only if there is a labeling $S:V(T)\to\{A,B,C,D\}$ such that $(T,S)$ belongs to the recursively constructed family $\mathcal{T}$ via these operations.

3. Constructive Characterizations and Structural Properties

For trees, UVR membership (i.e., $\gamma(G-v)=\gamma(G)$ for all $v$) is equivalent to

• Having a unique $\gamma_R$-function $f$ with $V_1^f=\emptyset$,
• $V_2^f$ (label-2 vertices) forming an independent set,
• Each label-2 vertex together with its two label-0 neighbors forming a private neighborhood of size 3.

Constructive labeling uses three statuses $\{A,B,C\}$ and a set of tree extension rules (e.g., attach 3-paths or 3-stars at special-status vertices): All UVR trees can be built from a labeled $K_{1,2}$ and recursive application of four attachment operations. In these trees, every B-vertex (which will be labeled 2 in the unique RDF) has exactly two A-neighbors, and no minimum RDF uses label 1 (Samodivkin, 2015).

For $\gamma_R$-excellent trees, the constructive process extends via additional statuses and attachments; in particular, every UVR tree is $\gamma_R$-excellent, with its vertex set partitioned into status C and D only (Samodivkin, 2016). In the broader graph context, UVR graphs satisfy

• $\gamma_R=2\gamma$,
• $\gamma_R\leq (2/3)n$,
• $b_R(G)\leq \delta(G)$, with $\delta(G)$ the minimum degree.

4. Comparative Classes and Robustness

Roman domination properties admit a taxonomy based on the stability of $\gamma_R$ under vertex and edge modifications (Samodivkin, 2017). The principal classes for $k=1$ changes are:

• CVR: $\gamma_R(G-v)<\gamma_R(G)$ for all $v$ (critical under vertex removal).
• UVR: $\gamma_R(G-v)=\gamma_R(G)$ for all $v$ (unchanged).
• CER/UEA: analogues for edge addition/deletion.

The intersection and containment relationships of these classes are extensive; for instance, CVR and UVR are disjoint, UVR is contained in UEA, and all graphs in UVR have label-1-free $\gamma_R$-functions. $\gamma_R$-excellent graphs generalize UVR robustness: They are resistant to reductions in Roman domination number under single-vertex deletions, as every vertex can be defended (appear with label 1 or 2) in some minimum RDF. UVR trees and their Roman domination analogues illustrate the sharp bounds and extremal resistance to perturbation: Bondage stability and unique RDFs within UVR directly transfer to the $\gamma_R$-excellent context (Samodivkin, 2016, Samodivkin, 2015).

5. Examples, Criticality, and Extremal Constructions

The following table summarizes several key examples from the literature on $\gamma_R$-excellent and UVR trees:

Graph/Tree	$\gamma_R$	$\gamma_R$-excellent?	Construction/Status
$K_2$	2	Yes	Both vertices labeled 1 or 2
$P_3$	2	No	Endpoints always 0
$C_{3k}$	$2k$	Yes	Efficient dominance, UVR
$P_{3k}$	$2k$	Yes	UVR, unique RDF
$P_4$	3	No	$\gamma_R$ drops on deletion

The recursive construction processes enable the generation of larger $\gamma_R$-excellent trees from seed configurations: For example, starting with $H_2$ (a framework of five vertices), repeated use of operations $O_3$ and $O_1$ produces increasingly complex $\gamma_R$-excellent trees, always preserving the status partition and properties required. This robustness and structural predictability suggest significant applications in network reliability and defense models, where stable resource allocations are crucial under small perturbations (Samodivkin, 2016).

6. Open Problems and Research Directions

Ongoing investigations include:

• Characterizing unicyclic and more complex graphs in $\mathcal{R}_{UVR}$.
• Determining extremal edge counts for given $(n,k)$ with $\gamma_R(G)=k$ in $\mathcal{R}_{UVR}$.
• Improving the upper bound $c<2/3$ so that $\gamma_R(G)\le c n$ for connected graphs with $\delta(G)\geq 3$.
• Developing reconfiguration graphs for $\gamma_R$ and related invariants, exploring connectivity and realizability aspects.
• Studying the strong Roman domination number $\gamma_{StR}(G)$, where additional requirements yield higher values (e.g., for trees, $\gamma_{StR}(T)\leq 6n/7$ with extremal constructions relying on rooted products of subdivided stars) (Alvarez-Ruiz et al., 2015).

A plausible implication is that further exploration of status-partition-based constructions and their algorithmic ramifications may yield efficient recognition and generation protocols for Roman domination-excellent and UVR graphs, relevant both in combinatorial optimization and dynamic network theory.