The stable maximum nullity of digraphs and $1$-DAGs

Given a digraph $D=(V,A)$ with vertex-set $V=\{1,\ldots,n\}$ and arc-set $A$, we denote by $Q(D)$ the set of all real $n\times n$ matrices $B=[b_{u,w}]$ with $b_{u,u}\not=0$ for all $u\in V$, $b_{u,w} \not= 0$ if $u\not=w$ and there is an arc from $u$ to $w$, and $b_{u,w}=0$ if $u\not=w$ and there is no arc from $u$ to $w$. We say that a matrix $B\in Q(D)$ has the Asymmetric Strong Arnold Property (ASAP) if $X\circ B = 0$, $X^T B = 0$, and $B X^T = 0$ implies $X=0$. We define the stable maximum nullity, $\overrightarrow{\nu}(D)$, of a digraph $D$ as the largest nullity of any matrix $A\in Q(D)$ that has the ASAP\@. We show that a digraph $D$ has $\overrightarrow{\nu}(D)\leq 1$ if and only $D$ and $\overleftarrow{D}$ a partial $1$-DAGs.