Packing and Squeezing Subgraphs into Planar Graphs

We consider the following problem: Given a set S of graphs, each of n vertices, construct an nvertex planar graph G containing all the graphs of S as subgraphs. We distinguish the variant in which any two graphs of S are required to have disjoint edges in G (known as ’packing’) from the variant in which distinct graphs of S can share edges in G (called ’squeezing’). About the packing variant we show that an arbitrary tree and an arbitrary spider tree can always be packed in a planar graph, improving in this way partial results recently given on this problem. Concerning the squeezing variant, we establish which classes of graphs can generally be squeezed in a planar graph, and which classes cannot.