Weighted Roman Domination Number

• Weighted Roman domination number is a graph invariant that generalizes traditional Roman domination to weighted graphs by assigning values from {0,1,2} while satisfying domination constraints.
• It establishes tight bounds in relation to the weighted domination number, with key equality cases observed in complete graphs, stars, and specific circulant weight patterns.
• Its duality with the weighted differential not only offers new insights in extremal graph theory but also facilitates modeling of biomolecular structures by reflecting critical vertex attributes.

The weighted Roman domination number generalizes the classical Roman domination number to weighted graphs, assigning a positive real weight to each vertex. The parameter measures the minimum total weighted cost of a function that assigns values from $\{0,1,2\}$ to vertices—subject to domination constraints inspired by the strategic placement of Roman legions on a network—while accounting for vertex weights. This concept extends and refines domination invariants in graph theory and finds natural applications in the modeling of biomolecular structures, where vertex weights reflect biologically meaningful attributes.

1. Formal Definitions and Key Parameters

Let $(G; w)$ denote a vertex-weighted graph where $G=(V,E)$ is a finite simple undirected graph and $w:V\rightarrow\mathbb{R}^{>0}$ assigns a strictly positive real weight to each vertex. The weight of a subset $S\subseteq V$ is $w(S)=\sum_{v\in S}w(v)$, and the total weight is $w(G)=w(V)$.

A weighted dominating set $D\subseteq V$ satisfies the property that every vertex $u\in V\setminus D$ has at least one neighbor in $D$. The weighted domination number is defined as

$$\gamma_w(G) = \min\{w(D): D\subseteq V \text{ and } D \text{ dominates } G\}.$$

A weighted Roman dominating function (wRDF) $f:V\rightarrow\{0,1,2\}$ satisfies that for every vertex $u$ with $f(u) = 0$, there exists a neighbor $v\in N(u)$ with $f(v)=2$. The weight of $f$ is $f(V) = \sum_{u\in V}f(u)w(u)$. The weighted Roman domination number is

$$\gamma_{wR}(G) = \min\{f(V): f \text{ is a wRDF on } (G;w)\}.$$

For weighted degree, $d_w(v) = w(N(v))/w(v)$, where $N(v)$ is the open neighborhood of $v$; the extremal weighted degrees are $\Delta_w = \max_{v\in V}d_w(v)$ and $\delta_w = \min_{v\in V}d_w(v)$. For $S \subseteq V$, let $$B(S) = \{x \in V\setminus S: x \text{ has a neighbor in } S\}$$. The weighted differential of $S$ is

$\partial(S) = w(B(S)) - w(S),$

and the differential of $(G; w)$ is $\partial(G) = \max_{S\subseteq V}\partial(S)$ (Cera et al., 27 Dec 2025).

2. Principal Bounds and Extremal Structures

Several foundational inequalities are established for $\gamma_{wR}(G)$ in relation to $\gamma_w(G)$:

• Domination–Roman Sandwich Theorem: For every $(G;w)$,

$\gamma_w(G) \le \gamma_{wR}(G) \le 2\gamma_w(G)$

The lower bound arises since any wRDF's support on $\{1,2\}$ is a dominating set, while the upper bound is realized by labeling a $\gamma_w$-set with $2$ and others with $0$ (Cera et al., 27 Dec 2025).

• Sharpness Conditions:
  • $\gamma_{wR}(G) = \gamma_w(G)$ if and only if $G$ is edgeless.
  • $\gamma_{wR}(G) = 2\gamma_w(G)$ for complete graphs or graphs where a $\gamma_w$-set is a distance $\ge 3$ set.
• Weighted $\Delta$-Bound:

$$\gamma_{wR}(G) \ge \left\lceil \frac{2w(G)}{\Delta_w + 1} \right\rceil$$

This bound is tight for weighted star graphs configured to equate the center’s weighted degree with $\Delta_w$.

• Trivial Weight Upper Bound:

$\gamma_{wR}(G) \le w(G)$

with equality if and only if every component is an isolated vertex or an isolated edge $\{x, y\}$ with $w(x)=w(y)$.

• Nordhaus–Gaddum Type Inequality: If $G$ and its complement $\overline{G}$ are nontrivial on $n \ge 3$ vertices,

$$4 \min_{v} w(v) \leq \gamma_{wR}(G) + \gamma_{wR}(\overline{G}) < w(G) + w(\overline{G})$$

3. Exact Values for Standard Graph Classes

Exact formulas are obtained for $\gamma_{wR}(G)$ in several canonical graph families:

Graph Family	$\gamma_{wR}(G)$	Notes
Complete Graph $(K_n; w)$	$2 \min_{v}w(v)$	Holds for all $n$ (Cera et al., 27 Dec 2025)
Star $(K_{1,t}; w)$	$\min\{2w(x_1),\, w(y_1) + w(Y)\}$	$x_1$ is the center; $Y$ is the leaf set
General $K_{s,t}; w$, $s \ge 2$	$$\min\{w(x_1)+w(X),\, w(y_1)+w(Y),\, 2[w(x_1)+w(y_1)]\}$$	$X, Y$ color classes ordered by weights
Cycle $(C_n; w)$	$\leq (1 - k/n)w(C_n)$, $k = \lfloor n/3 \rfloor$	Equality if $w$ is constant, $n=3k+\ell$, $\ell=1,2$

For nonuniform weights or $n$ divisible by $3$, various circulant weight patterns realize the tight upper bound, as the associated linear system admits infinitely many solutions (Cera et al., 27 Dec 2025).

4. Realizability and Extremal Constructions

The extremal realizations corresponding to the established bounds are structurally characterized:

• Equality $\gamma_{wR}=2\gamma_w$ is achieved in complete graphs and graphs where a $\gamma_w$-set is isolated at distance at least $3$.
• Equality $\gamma_{wR} = \lceil 2 w(G)/(\Delta_w + 1)\rceil$ is realized by specifically weighted stars with the central vertex’s weighted degree matching $\Delta_w$.
• Equality $\gamma_{wR} = w(G)$ holds only for graphs where each connected component is either an isolated vertex or an isolated edge with balanced weights (Cera et al., 27 Dec 2025).

5. Equivalence with the Weighted Differential

A central result demonstrates that the weighted Roman domination number is dual to the graph’s maximal weighted differential: $\gamma_{wR}(G) = w(G) - \partial(G)$ where $\partial(G) = \max_{S\subseteq V}[w(B(S)) - w(S)]$. For an optimal wRDF $f$ with $V_2 = S$, the vertices labeled $2$ correspond to $S$, those labeled $0$ correspond to $B(S)$, and those labeled $1$ to $V\setminus (S \cup B(S))$. The construction yields $$f(V) = w(G) - \left[w(B(S)) - w(S)\right] \ge w(G) - \partial(G)$$, while a set $S^*$ achieving $\partial(G)$ immediately induces a wRDF of weight $w(G) - \partial(G)$. This one-to-one correspondence underpins applications in extremal combinatorics and duality theory in domination (Cera et al., 27 Dec 2025).

6. Applications and Significance

Weighted Roman domination numbers extend traditional domination parameters to encompass weighted models, providing greater flexibility for applications requiring variable importance of vertices—such as the modeling of biomolecular structures in bioinformatics, where vertex weights encode structural or functional information. This weighted framework is essential for accurate representation and optimization in biological and computational settings, offering sharper and more relevant bounds than unweighted analogues.

The equivalence between weighted Roman domination and the weighted differential links the parameter to maximization problems over reweighted neighborhoods, opening avenues for duality theory and extremal analysis in discrete mathematics and algorithmic graph theory (Cera et al., 27 Dec 2025).