  Eternal domination is a dynamic process by which a graph is protected from an
infinite sequence of vertex intrusions. In eternal $k$-domination, guards
initially occupy the vertices of a $k$-dominating set. After a vertex is
attacked, guards "defend" by each move up to distance $k$ to form a
$k$-dominating set containing the attacked vertex. The eternal $k$-domination
number of a graph is the minimum number of guards needed to defend against any
sequence of attacks. The process is well-studied for the $k=1$ situation and we
introduce eternal $k$-domination for $k > 1$.
  Determining if a given set is an eternal $k$-domination set is in EXP, and in
this paper we provide a number of results for paths and cycles, and relate this
parameter to graph powers and domination in general. For trees we utilize
decomposition arguments to bound the eternal $k$-domination numbers, and solve
the problem entirely in the case of perfect $m$-ary trees.
