What question did this study set out to answer?

The paper aims to investigate the Roman domination number in the context of zero-divisor graphs of commutative rings with finite domination numbers.

June 29, 2026

On the Roman domination number of the zero-divisor graph of a commutative ring

Puntos clave

The paper aims to investigate the Roman domination number in the context of zero-divisor graphs of commutative rings with finite domination numbers.
Define the structure of zero-divisor graphs based on commutative rings with identity and nonzero zero-divisors.
Introduce Roman dominating functions and analyze their properties within these graphs.
Restrict to functions that yield finitely many vertices with specific dominance properties.
Determine conditions under which the Roman domination number remains finite.
Provide new results related to the weight of different Roman dominating functions on zero-divisor graphs.
Identify specific structures within zero-divisor graphs that affect the domination number.

Resumen

The rings considered in this paper are commutative with identity that admit at least one nonzero zero-divisor. Let Formula: see text be a ring and let Formula: see text denote the set of all zero-divisors of Formula: see text. Let us denote Formula: see text by Formula: see text. The zero-divisor graph of Formula: see text, denoted by Formula: see text, is an undirected graph such that its vertex set is Formula: see text and distinct vertices Formula: see text and Formula: see text are adjacent in Formula: see text if and only if Formula: see text. A Roman dominating function on a finite graph Formula: see text is a function Formula: see text that satisfies the following property: any Formula: see text for which Formula: see text must be adjacent in Formula: see text to at least one vertex Formula: see text for which Formula: see text. The weight of a Roman dominating function Formula: see text is the value Formula: see text. The minimum weight of a Roman dominating function on Formula: see text is called the Roman domination number of Formula: see text, denoted by Formula: see text. As Formula: see text need not be finite, we restrict ourselves to Roman dominating functions Formula: see text on Formula: see text such that there are only finitely many Formula: see text with Formula: see text or Formula: see text. This paper aims to discuss some results on the Roman domination number of Formula: see text with the assumption that the domination number of Formula: see text is finite.

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