Beyond the pseudoforest strong Nine Dragon Tree Theorem

The pseudoforest version of the Strong Nine Dragon Tree Conjecture states that if a graph $G$ has maximum average degree $mad\left(G\right)=2{max}_{H\subseteq G}\frac{e\left(H\right)}{v\left(H\right)}$ at most $2(k+\frac{d}{d+k+1})$, then it has a decomposition into $k+1$ pseudoforests where in one pseudoforest $F$ the components of $F$ have at most $d$ edges. This was proven in 2020 in Grout and Moore (2020). We strengthen this theorem by showing that we can find such a decomposition where additionally $F$ is acyclic, the diameter of the components of $F$ is at most $2\ell +2$, where $\ell =\left(\frac{d-1}{k+1}\right)$, and at most $2\ell +1$ if $d\equiv 1mod(k+1)$. Furthermore, for any component $K$ of $F$ and any $z\in N$, we have $diam\left(K\right)\le 2z$ if $e\left(K\right)\ge d-z(k-1)+1$. We also show that both diameter bounds are best possible as an extension for both the Strong Nine Dragon Tree Conjecture for pseudoforests and its original conjecture for forests. In fact, they are still optimal even if we only enforce $F$ to have any constant maximum degree, instead of enforcing every component of $F$ to have at most $d$ edges.