On the number of terms in the middle of almost split sequences over 
tame algebras
J. A. de la Pena and M. Takane
Trans. Amer. Math. Soc. 351 (1999), 3857-3868.

Abstract. Let $A$ be a finite dimensional tame algebra over an algebraically 
closed field $k$. It has been conjectured that any almost split sequence 
$0 \rightarrow X \rightarrow \bigoplus_{i=1}^n Y_i \rightarrow Z \rightarrow 0$
with $Y_i$ indecomposable modules has $n \le 5$ and in case $n=5$, then 
exactly one of the $Y_i$ is a projective-injective module. In this
work we show this conjecture in case all the $Y_i$ are directing modules, that 
is, there are no cycles of non-zero, non-iso maps $Y_i = M_1
\rightarrow M_2 \rightarrow \dots \rightarrow M_s = Y_i$
between indecomposable $A$-modules. In case, $Y_1$ and $Y_2$ are isomorphic, 
we show that $n \le 3$ and give precise information on the structure of $A$.