  Let $G$ be a connected graph. A pebbling move is defined as taking two
pebbles from one vertex and placing one pebble to an adjacent vertex and
throwing away the other pebble. The non-split domination cover pebbling number,
$\psi_{ns}(G)$, of a graph $G$ is the minimum of pebbles that must be placed on
$V(G)$ such that after a sequence of pebbling moves, the set of vertices with a
pebble forms a non-split dominating set of $G$, regardless of the initial
configuration of pebbles. We discuss some basic results, NP-completeness of
non-split domination number, and determine $\psi_{ns}$ for some families of
Middle graphs.
