\section{Extension via a stated conservative labeling theorem}\label{sec:extension}
The previous results do not use the external theorem discussed here. Licona and Tey~\cite[Theorem 4.17, p.~23]{licona2024} state that every snowflake with $M$ edges has a conservative labeling if $M\equiv0,3\pmod4$, and a near-conservative labeling otherwise. These are bijective edge labelings by $[M]$ and $[M-1]\cup\{M+1\}$, respectively, with zero signed incident sum at every vertex of degree at least three. A snowflake is obtained by identifying one leaf from each of a collection of disjoint stars~\cite[Definition 4.1, p.~13]{licona2024}.

\begin{proposition}[Conditional on the stated snowflake theorem]\label{prop:extension}
Assume Theorem 4.17 of~\cite{licona2024}. Let $s\geq1$, $r\geq0$, and $a_i,b_i\geq1$, and put $M=\sum_{i=1}^{s}(a_i+b_i)$. Then
\[
\left(\bigsqcup_{i=1}^{s}K_{a_i,b_i}\right)\sq rK_1
\text{ is DDMO}
\quad\Longleftrightarrow\quad
\begin{cases}
a_i,b_i\geq3&\text{for every }i,\\
r\geq1\text{ or }M\equiv0,3\pmod4.&
\end{cases}
\]
Here $M$ counts the nonisolated vertices, not the total number of vertices.
\end{proposition}
\begin{proof}
The degree-one and degree-two obstructions require all part sizes to be at least three. If $r=0$, Lemma~\ref{lem:bipartite} makes every part sum even, so $M(M+1)/2$ is even, equivalently $M\equiv0,3\pmod4$.

For sufficiency, create one star with $d$ edges for each prescribed part size $d$. If there are at least three stars, identify one leaf of each to form a snowflake and apply the assumed theorem. Each original star center still has degree $d\geq3$, so its incident edge labels form a signed-zero block of the required size. Splitting the identified vertex back into leaves leaves these blocks unchanged. Lemma~\ref{lem:bipartite} now supplies the orientations.

For exactly two stars, if some size $d\geq6$, replace it temporarily by two sizes $3$ and $d-3$. The preceding construction applies to the resulting three stars. Merge the two corresponding signed-zero blocks to recover size $d$; their signed sum remains zero. To avoid depending on a degree-two-center boundary convention for the external theorem, the six remaining cases are supplied directly below. Each row gives two signed-zero blocks; the absolute values cover $[M]$ in the ordinary cases and $[M-1]\cup\{M+1\}$ otherwise:
\[
\begin{array}{c|l|l}
(a,b)&\text{first signed block}&\text{second signed block}\\\hline
(3,3)&(1,3,-4)&(2,5,-7)\\
(3,4)&(1,2,-3)&(4,-5,-6,7)\\
(3,5)&(1,3,-4)&(2,5,-6,7,-8)\\
(4,4)&(1,-2,-3,4)&(5,-6,-7,8)\\
(4,5)&(1,-2,-3,4)&(-5,-6,-7,8,10)\\
(5,5)&(-1,2,-3,-4,6)&(-5,-7,-8,9,11).
\end{array}
\]
In the ordinary cases, any isolates receive labels $M+1,\ldots,M+r$. In the near cases, give one isolate label $M$ and any further isolates labels $M+2,\ldots,M+r$.
\end{proof}

\paragraph{Why the dependence is explicit.}
The printed Lemma 3.17 of~\cite[p.~13]{licona2024} has a parity inconsistency: $n_1=3$, $n_2=1$, and $k=t=0$ satisfy its stated condition on $n=n_1+n_2$, but would give signed-zero blocks with absolute values partitioning $[14]$. Their total absolute sum would be even, whereas $1+\cdots+14=105$. We have not fully verified the external proof or established a counterexample to its Theorem 4.17. Accordingly Proposition~\ref{prop:extension} is a conditional reduction, not an independently established general classification in this note.

Licona and Tey's introduction~\cite[p.~3]{licona2024} attributes the related galaxy result to Goldfeder and Tey (2018), their reference 13. We do not claim the partition principle as new. The freely available 2024 paper is directly cited here; we did not obtain a freely readable full text of that older dependency.