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The pseudoforest version of the Strong Nine Dragon Tree Conjecture statesthat if a graph $G$ has maximum average degree $\text{mad}(G) = 2 \max_{H\subseteq G} \frac{e(G)}{v(G)}$ at most $2(k + \frac{d}{k+d+1})$, then it has adecomposition into $k+1$ pseudoforests where in one pseudoforest $F$ thecomponents of $F$ have at most $d$ edges. This was proven in 2020. Westrengthen this theorem by showing that we can find such a decomposition whereadditionally $F$ is acyclic, the diameter of the components of $F$ is at most$2\ell + 2$, where $\ell = \lfloor\frac{d-1}{k+1} \rfloor$, and at most $2\ell+ 1$ if $d \equiv 1 \bmod k+1$. Furthermore, for any component $K$ of $F$ andany $z \in \mathbb N$, we have $diam(K) \leq 2z$ if $e(K) \geq d - z(k-1) + 1$.We also show that both diameter bounds are best possible as an extension forboth the Strong Nine Dragon Tree Conjecture for pseudoforests and its originalconjecture for forests. In fact, they are still optimal even if we only enforce$F$ to have any constant maximum degree, instead of enforcing every componentof $F$ to have at most $d$ edges.

Beyond the Pseudoforest Strong Nine Dragon Tree Theorem