I am aware that Hall's Marriage theorem for complete matching goes like "A bipartite graph $G$ with bipartition $(V_1, V_2)$ has a complete matching from $V_1$ to $V_2$ if and only if $$ |N(A)| \geq |A|, \forall A \subseteq V_1$$
I want to know in which cases does an equality hold, i.e. $$ |N(A)| = |A|, \forall A \subseteq V_1 $$
Any help is greatly appreciated.