  Based on the history that the Emperor Constantine decreed that any undefended
place (with no legions) of the Roman Empire must be protected by a "stronger"
neighbor place (having two legions), a graph theoretical model called Roman
domination in graphs was described. A Roman dominating function for a graph
$G=(V,E)$, is a function $f:V\rightarrow \{0,1,2\}$ such that every vertex $v$
with $f(v)=0$ has at least a neighbor $w$ in $G$ for which $f(w)=2$. The Roman
domination number of a graph is the minimum weight, $\sum_{v\in V}f(v)$, of a
Roman dominating function.
  In this paper we initiate the study of a new parameter related to Roman
domination, which we call strong Roman domination number and denote it by
$\gamma_{StR}(G)$. We approach the problem of a Roman domination-type defensive
strategy under multiple simultaneous attacks and begin with the study of
several mathematical properties of this invariant. In particular, we first show
that the decision problem regarding the computation of the strong Roman
domination number is NP-complete, even when restricted to bipartite graphs. We
obtain several bounds on such a parameter and give some realizability results
for it. Moreover, we prove that for any tree $T$ of order $n\ge 3$,
$\gamma_{StR}(T)\le 6n/7$ and characterize all extremal trees.
