On the $nk-attack Roman Dominating Number of a Graph

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. We consider the $k$-attack Roman Domination, particularly focusing on 2-attack Roman Domination. A Roman Dominating function of $G$ is a $k$-attack Roman Dominating function of $G$ if for all $j\leq k$, 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 $k$-attack Roman Dominating number of $G, \gkaRD{G}$, is the minimum total weight over all possible $k$-attack Roman Dominating functions. We find $\gtaRD{G}$ for particular graph class, discuss properties of $k$-attack Roman Domination, and make several connections with other domination ideas.