Roman domination in weighted graphs

TL;DR

This paper introduces the concept of weighted Roman domination in graphs, establishing bounds, exact values for specific graph families, and linking it to the graph differential, with applications in bioinformatics.

Contribution

It is the first to study weighted Roman domination in graphs, providing bounds, realizability results, and connections to graph differential.

Findings

Established bounds for weighted Roman domination number

Determined exact values for well-known graph families

Proved equivalence with graph differential

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 $\gamma_{R}(G)$. 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 $\gamma_{wR}(G)$. 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.