Lengths of words in transformation semigroups generated by digraphs

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.