Packing and Squeezing Subgraphs into Planar Graphs

Abstract

We consider the following problem: Given a set S of graphs, each of n vertices, construct an n-vertex 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 consider some important classes of graphs, like paths, trees, outerplanar graphs, etc. and establish positive and negative results.