  The Roman Dominating number is a widely studied variant of the dominating
number on graphs. Given a graph $G=(V,E)$, the dominating number of a graph is
the minimum size of a vertex set, $V' \subseteq V$, so that every vertex in the
graph is either in $V'$ or is adjacent to a vertex in $V'$. A Roman Dominating
function of $G$ is defined as $f:V \rightarrow \{0,1,2\}$ such that every
vertex with a label of 0 in $G$ is adjacent to a vertex with a label of 2. The
Roman Dominating number of a graph is the minimum total weight over all
possible Roman Dominating functions. In this paper, we analyze a new variant:
$n$-attack Roman Domination, particularly focusing on 2-attack Roman Domination
$(n=2)$. An $n$-attack Roman Dominating function of $G$ is defined similarly to
the Roman Dominating function with the additional condition that, for any
$j\leq n$, any subset $S$ of $j$ vertices all with label 0 must have at least
$j$ vertices with label 2 in the open neighborhood of $S$. The $n$-attack Roman
Dominating number is the minimum total weight over all possible $n$-attack
Roman Dominating functions. We introduce properties as well as an algorithm to
find the 2-attack Roman Domination number. We also consider how to place
dominating vertices if you are only allowed a finite number of them through the
help of a python program. Finally, we discuss the 2-attack Roman Dominating
number of infinite regular graphs that tile the plane. We conclude with open
questions and possible ways to extend these results to the general $n$-attack
case.
