<p>We introduce the Edge-Cost double Roman domination problem, a generalization of the standard double Roman domination (DRD) problem. While the standard DRD problem focuses only on the number of units placed on vertices of an undirected graph to guarantee “double Roman” level of security, Edge-Cost double Roman domination additionally incorporates the cost of moving these units along edges in case of an emergency. More specifically, edges are assigned numerical costs representing the effort, risk, or delay of moving a single unit between locations. These costs can reflect time delays, fuel consumption, or other operational constraints, making the problem more realistic for applications in logistics, disaster response, and communication networks. We study two different variants: Sum-Cost, where the total cost is the sum of deployed units and the cumulative travel cost of all assistance needed, and Worst-Case, where the total cost is the sum of deployed units and the maximum single travel cost required to provide help. These two variants capture different operational priorities: efficiency versus robustness. For each variant, we give formal definitions and prove NP-hardness on general graphs as well as on many graph families. Also, we design separate polynomial time algorithms for solving each variant on trees. Furthermore, we prove that for the Worst-Case variant there exists an optimal solution in which no vertex is assigned the value 1.</p>