  We study the relationship between the eternal domination number of a graph
and its clique covering number using both large-scale computation and analytic
methods. In doing so, we answer two open questions of Klostermeyer and
Mynhardt. We show that the smallest graph having its eternal domination number
less than its clique covering number has $10$ vertices. We determine the
complete set of $10$-vertex and $11$-vertex graphs having eternal domination
numbers less than their clique covering numbers. We show that the smallest
triangle-free graph with this property has order $13$, as does the smallest
circulant graph. We describe a method to generate an infinite family of
triangle-free graphs and an infinite family of circulant graphs with eternal
domination numbers less than their clique covering numbers. We also consider
planar graphs and cubic graphs. Finally, we show that for any integer $k \geq
2$ there exist infinitely many graphs having domination number and eternal
domination number equal to $k$ containing dominating sets which are not eternal
dominating sets.
