An edge-centric perspective of Roman domination in fuzzy graphs through strong neighborhoods

TL;DR

This paper extends Roman domination concepts to fuzzy graphs by introducing the strong-neighbors Roman domination number, analyzing its properties, bounds, and specific cases like fuzzy cycles and paths.

Contribution

It introduces the strong-neighbors Roman domination number for fuzzy graphs and explores its properties, bounds, and specific graph classes, extending classical domination theory.

Findings

Determined the strong-neighbors Roman domination number for complete fuzzy graphs.

Established bounds for the strong-neighbors Roman domination number.

Characterized fuzzy graphs with extreme domination values, including cycles and paths.

Abstract

This work is related to the extension of the well-known problem of Roman domination in graph theory to fuzzy graphs. A variety of approaches have been used to explore the concept of domination in fuzzy graphs. This study uses the concept of strong domination, considering the weights of the strong edges. We introduce the strong-neighbors Roman domination number of a fuzzy graph and establish some correlations with the Roman domination in graphs. The strong-neighbors Roman domination number is determined for specific fuzzy graphs, including complete and complete bipartite fuzzy graphs. Besides, several general bounds are given. In addition, we characterize the fuzzy graphs that reach the extreme values with particular attention to fuzzy strong cycles and paths.