in the directed cycle is not unique. Since the size of the dicycle in elements of $\mathcal{T}$ is restricted, we are able to easily list, in Figure \[fig:T\], the four members of $\mathcal{T}$.

\[T\] If $\overrightarrow{D} \in \mathcal{T}$, then $obt(\overrightarrow{D}) > 1$.

![Members of $\mathcal{T}$[]{data-label="fig:T"}](tripod_class.eps)

To introduce the last class, $\mathcal{R}$, of strictly uni-dicyclic graphs, we first define an oriented tree, called an *antler*, using the oriented paths $S^+, S^-,$ and a standard $j$-dipath $\overrightarrow{P_j}$. Let $s^+, s^-$ be the vertices of degree two in $S^+$, $S^-$, respectively. For an integer $j \geq 2$, we define a *positive $j$-antler*, $\overrightarrow{A}_j^+ := S^+(s^+) +_1 \overrightarrow{P_j}(j)$; we also define the positive $1$-antler $A^+_1 := S^+$. We define a *negative $j$-antler* $A^-_j$ to be the converse, $(A^+_j)^*$, of a positive $j$-antler. $A^+_3$ and $A^-_4$ can be seen in Figure \[fig:antler\]. It is important to note that if an oriented tree contains *no* positive antler, it is a negative fountain tree, and similarly, if an oriented tree contains *no* negative antler, it is a positive fountain tree.

![$A^+_3$ and $A^-_4$[]{data-label="fig:antler"}](antlers.eps)

We now define $\mathcal{R}$, using a positive antler $A^+_j$, a negative antler $A^-_k$, and the standard $n$-dicycle, $\overrightarrow{D_n}$. In $\overrightarrow{D_n}$, we choose an arbitrary 