article published yes Sebastian Mies author Benjamin Moore author 6dc1a1be-bf1c-11ed-8d2b-d044840f49d6 Evelyne Smith-Roberge author MaKw department The pseudoforest version of the Strong Nine Dragon Tree Conjecture states that if a graph G has maximum average degree mad(G) = 2 maxH⊆G e(H)/v(H) at most 2(k + 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 2ℓ + 2, where ℓ =⌊d−1/k+1⌋, and at most 2ℓ + 1 if d ≡ 1 mod (k + 1). Furthermore, for any component K of F and any z ∈ N, we have diam(K) ≤ 2z if e(K) ≥ 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. https://research-explorer.ista.ac.at/download/20320/20913/2025_EuropJournCombinatorics_Mies.pdf application/pdfno Elsevier2025 eng 0195-6698 2310.00931 00152976930000210.1016/j.ejc.2025.104214 13012 S. Mies, B. Moore, and E. Smith-Roberge, “Beyond the pseudoforest strong Nine Dragon Tree theorem,” <i>European Journal of Combinatorics</i>, vol. 130, no. 12. Elsevier, 2025. Mies S, Moore B, Smith-Roberge E. 2025. Beyond the pseudoforest strong Nine Dragon Tree theorem. European Journal of Combinatorics. 130(12), 104214. Mies, Sebastian, et al. “Beyond the Pseudoforest Strong Nine Dragon Tree Theorem.” <i>European Journal of Combinatorics</i>, vol. 130, no. 12, 104214, Elsevier, 2025, doi:<a href="https://doi.org/10.1016/j.ejc.2025.104214">10.1016/j.ejc.2025.104214</a>. Mies S, Moore B, Smith-Roberge E. Beyond the pseudoforest strong Nine Dragon Tree theorem. <i>European Journal of Combinatorics</i>. 2025;130(12). doi:<a href="https://doi.org/10.1016/j.ejc.2025.104214">10.1016/j.ejc.2025.104214</a> Mies, Sebastian, Benjamin Moore, and Evelyne Smith-Roberge. “Beyond the Pseudoforest Strong Nine Dragon Tree Theorem.” <i>European Journal of Combinatorics</i>. Elsevier, 2025. <a href="https://doi.org/10.1016/j.ejc.2025.104214">https://doi.org/10.1016/j.ejc.2025.104214</a>. Mies, S., Moore, B., &#38; Smith-Roberge, E. (2025). Beyond the pseudoforest strong Nine Dragon Tree theorem. <i>European Journal of Combinatorics</i>. Elsevier. <a href="https://doi.org/10.1016/j.ejc.2025.104214">https://doi.org/10.1016/j.ejc.2025.104214</a> S. Mies, B. Moore, E. Smith-Roberge, European Journal of Combinatorics 130 (2025). 203202025-09-10T05:36:50Z2025-12-30T10:19:10Z