  Given a simple digraph $D$ on $n$ vertices (with $n\ge2$), there is a natural
construction of a semigroup $\langle D\rangle$ associated with $D$. For any
edge $(a,b)$ of $D$, let $a\to b$ be the idempotent of defect $1$ mapping $a$
to $b$ and fixing all vertices other than $a$; then define $\langle D\rangle$
to be the semigroup $\langle a\to b:(a,b)\in E(D)\rangle$. For $\alpha \in
\langle D \rangle$, let $\ell(D,\alpha)$ be the minimal length of a word in
$E(D)$ expressing $\alpha$. When $D=K_n$ is the complete undirected graph,
Howie and Iwahori, independently, obtained a formula to calculate
$\ell(K_n,\alpha)$, for any $\alpha \in \langle K_n \rangle = \text{Sing}_n$;
however, no analogous nontrivial results are known when $D \neq K_n$. In this
paper, we characterise all simple digraphs $D$ such that either
$\ell(D,\alpha)$ is equal to Howie-Iwahori's formula for all $\alpha \in
\langle D \rangle$, or $\ell(D,\alpha) = n - \text{fix}(\alpha)$ for all
$\alpha \in \langle D \rangle$, or $\ell(D,\alpha) = n - \text{rk}(\alpha)$ for
all $\alpha \in \langle D \rangle$. When $D$ is an acyclic digraph and $\alpha
\in \langle D \rangle$, we find a tight upper bound for $\ell(D,\alpha)$.
Finally, we study the case when $D$ is a strong tournament (which corresponds
to a smallest generating set of idempotents of defect $1$ of $\text{Sing}_n$),
and we propose some conjectures.
