© 2024 World Scientific and Engineering Academy and Society. All rights reserved.Domination in graphs is a widely studied field, where many different definitions have been introduced in the last years to respond to different network requirements. This paper presents a new dominating parameter based on the well-known strong Roman domination model. Given a positive integer p, we call a p-strong Roman domination function (p-StRDF) in a graph G to a function f : V (G) ? {0, 1, 2, . . ., l ?+ppm } having the property that if f(v) = 0, then there is a vertex u ? N(v) such that f(u) ? 1 + l |B0?pN(u)|m , where B0 is the set of vertices with label 0. The p-strong Roman domination number ?StRp (G) is the minimum weight (sum of labels) of a p-StRDF on G. We study the NP-completeness of the p-StRD-problem, we also provide general and tight upper and lower bounds depending on several classical invariants of the graph and, finally, we determine the exact values for some families of graphs.