A note on a Brooks' type theorem for DP-coloring

Dvo\v{r}'{a}k and Postle \cite{DP} introduced a $\mathit{\text{DP-coloring}}$ of a simple graph as a generalization of a list-coloring. They proved a Brooks’ type theorem for a DP-coloring, and Bernshteyn, Kostochka and Pron \cite{BKP} extended it to a DP-coloring of multigraphs. However, detailed structure when a multigraph does not admit a DP-coloring was not specified in \cite{BKP}. In this note, we make this point clear and give the complete structure. This is also motivated by the relation to signed coloring of signed graphs.