Optimizing Emergency-Station Placement through Roman and Double Roman Domination in Zero-Divisor Graphs of Finite Rings

Zero-divisor graphs provide a powerful bridge between commutative algebra and graph theory by encoding annihilation relationships among zero divisors of a finite ring. While parameters such as connectivity, chromatic number, and classical domination have been extensively explored, Roman domination and double Roman domination in zero-divisor graphs remain largely uncharacterized. These parameters model optimal resource allocation under different levels of protection and redundancy. This study aims to compute and analyze the Roman domination number and the double Roman domination number for key families of finite commutative rings and to establish structural relationships between algebraic properties of rings and optimal domination behavior.

Methods:
For each selected ring family, zero-divisor graphs are constructed from annihilation relations. Roman and double Roman domination are formulated as constrained optimization problems, supported by algebraic–graph theoretic lemmas, degree-based bounds, and annihilator structure analysis. Exact values are obtained using integer linear programming (ILP), while large graphs are studied using heuristic algorithms informed by neighborhood structures and annihilator classes.

Results:
Preliminary analyses reveal that vertices with maximal annihilator degree play a central role in optimal Roman and double Roman labelings. Structural features such as star-like patterns, complete subgraphs, and decompositions from the Chinese Remainder Theorem significantly reduce domination cost. Across ring families, distinct algebraic characteristics—particularly annihilator chains and idempotent behavior—strongly influence domination numbers.

Conclusion:
This work establishes new theoretical bounds, exact values, and structural characterizations of Roman and double Roman domination in zero-divisor graphs. These findings also support an application framework interpreting domination assignments as minimum-cost emergency-station placement strategies, demonstrating the practical relevance of the studied parameters.