Subgraph-universal planar graphs for trees

We show that there exists an outerplanar graph on O (n^c) vertices for c = ₂ (3+10) 2. 623 that contains every tree on n vertices as a subgraph. This extends a result of Chung and Graham from 1983 who showed that there exist (non-planar) n-vertex graphs with O (n n) edges that contain all trees on n vertices as subgraphs and a result from Gol'dberg and Livshits from 1968 who showed that there exists a universal tree for n-vertex trees on n^O ( (n) ) vertices. Furthermore, we determine the number of vertices needed in the worst case for a planar graph to contain three given trees as subgraph to be on the order of 32n, even if the three trees are caterpillars. This answers a question recently posed by Alecu et al. in 2024. Lastly, we investigate (outer) planar graphs containing all (outer) planar graphs as subgraph, determining exponential lower bounds in both cases. We also construct a planar graph on n^O ( (n) ) vertices containing all n-vertex outerplanar graphs as subgraphs.