Roman domination in weighted graphs

TL;DR

The paper extends Roman domination to vertex-weighted graphs by defining the weighted Roman domination number $\gamma_{wR}(G)$ and exploring its theoretical properties. It derives general bounds $\gamma_w(G) \le \gamma_{wR}(G) \le 2 \gamma_w(G)$ and a lower bound $\gamma_{wR}(G) \ge \left\lceil \frac{2 w(G)}{\Delta_w+1} \right\rceil$, and analyzes equality conditions. Exact values are obtained for several graph families, including complete graphs where $\gamma_{wR}(K_n)=2 \min\{ w(v) \}$ and complete bipartite graphs with a specific formula, along with cycle bounds that depend on $n \bmod 3$. A key contribution is the proven equivalence $\gamma_{wR}(G)= w(G) - \partial(G)$ with the differential, linking domination concepts to a diffusion-like measure and enabling new interpretations and applications.

Abstract

A Roman dominating function for a (non-weighted) graph $G=(V,E)$, is a function $f:V\rightarrow \{0,1,2\}$ such that every vertex $u\in V$ with $f(u)=0$ has at least {one} neighbor $v\in V$ such that $f(v)=2$. The minimum weight $\sum_{v\in V}f(v)$ of a Roman {dominating function} $f$ on $G$ is called the Roman domination number of $G$ and is denoted by $γ_{R}(G)$. A graph {$G= (V,E)$} together with a positive real-valued weight-function $w:V\rightarrow \mathbf{R}^{>0}$ is called a {\it weighted graph} and is denoted by $(G;w)$. The minimum weight $\sum_{v\in V}f(v)w(v)$ of a Roman {dominating function} $f$ on $G$ is called the weighted Roman domination number of $G$ and is denoted by $γ_{wR}(G)$. The domination and Roman domination numbers of unweighted graphs have been extensively studied, particularly for their applications in bioinformatics and computational biology. However, graphs used to model biomolecular structures often require weights to be biologically meaningful. In this paper, we initiate the study of the weighted Roman domination number in weighted graphs. We first establish several bounds for this parameter and present various realizability results. Furthermore, we determine the exact values for several well-known graph families and demonstrate an equivalence between the weighted Roman domination number and the differential of a weighted graph.

Figures (3)

• Figure 1: An example for which the upper bound of Theorem \ref{['romcot']} is reached.
• Figure 2: Three different strategies depending on the distribution of vertex weights.
• Figure 3: Two examples of weighted hexagons where the equality in Theorem \ref{['cyclecota']} holds.

Theorems & Definitions (25)

• Theorem 3.1
• proof
• Remark 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• Proposition 3.5
• proof
• Lemma 3.6