  We study the directed Abelian sandpile model on a square lattice, with $K$
downward neighbors per site, $K > 2$. The $K=3$ case is solved exactly, which
extends the earlier known solution for the $K=2$ case. For $K>2$, the avalanche
clusters can have holes and side-branches and are thus qualitatively different
from the $K=2$ case where avalance clusters are compact. However, we find that
the critical exponents for $K>2$ are identical with those for the $K=2$ case,
and the large scale structure of the avalanches for $K>2$ tends to the $K=2$
case.
