Strong Equality of Roman and Weak Roman Domination in Trees

TL;DR

This paper characterizes trees where the minimum weak Roman domination functions are always also Roman dominating functions, revealing structural properties through five extension operations.

Contribution

It provides a constructive characterization of such trees, establishing when weak and strong Roman domination numbers coincide.

Findings

Identifies five extension operations that characterize these trees.

Reveals structural properties of trees with equal Roman and weak Roman domination numbers.

Provides a basis for recognizing such trees in graph theory applications.

Abstract

We provide a constructive characterization of the trees for which the Roman domination number strongly equals the weak Roman domination number, that is, for which every weak Roman dominating function of minimum weight is a Roman dominating function. Our characterization is based on five simple extension operations, and reveals several structural properties of these trees.