  The concept of Roman domination has been a subject of intrigue for more than
two decades with the fundamental Roman domination problem standing out as one
of the most significant challenges in this field. This article studies a
practically motivated generalization of this problem, known as the k-strong
Roman domination. In this variation, defenders within a network are tasked with
safeguarding any k vertices simultaneously, under multiple attacks. The
objective is to find a feasible mapping that assigns an (integer) weight to
each vertex of the input graph with a minimum sum of weights across all
vertices. A function is considered feasible if any non-defended vertex, i.e.
one labeled by zero, is protected by at least one of its neighboring vertices
labeled by at least two. Furthermore, each defender ensures the safety of a
non-defended vertex by imparting a value of one to it while always retaining a
one for themselves. To the best of our knowledge, this paper represents the
first theoretical study on this problem. The study presents results for general
graphs, establishes connections between the problem at hand and other
domination problems, and provides exact values and bounds for specific graph
classes, including complete graphs, paths, cycles, complete bipartite graphs,
grids, and a few selected classes of convex polytopes. Additionally, an
attainable lower bound for general cubic graphs is provided.
