Characterization of strongly ${\mathbb{Z}}_{\mathrm{\ell }}$-connected graphs of small order

A graph is strongly ${\mathbb{Z}}_{\mathrm{\ell }}$-connected if for each boundary function $\beta :V(G)\mapsto {\mathbb{Z}}_{\mathrm{\ell }}$ with $\beta (v)\equiv d(v)(\mathrm{m}\mathrm{o}\mathrm{d}2)$ for every vertex $v$ and ${\sum }_{v\in V(G)}\beta (v)\equiv 0(\mathrm{m}\mathrm{o}\mathrm{d}2\mathrm{\ell })$, there exists an orientation $D$ of $G$ such that $d_D^{+}(v)-d_D^{-}(v)\equiv \beta (v)(\mathrm{m}\mathrm{o}\mathrm{d}2\mathrm{\ell })$ for each $v\in V(G)$. This is a useful notion for studying circular flows of graphs. This note presents a fully self-contained, manual proof of a characterization of $4$-vertex strongly ${\mathbb{Z}}_{\mathrm{\ell }}$-connected graphs for any integer $\mathrm{\ell }\ge 2$, which will be used in our further study in this topic.