Embedding Complete Binary Trees into Star and Pancake Graphs

Abstract.

Let G and H be two simple undirected graphs. An embedding of the graph G into the graph H is an injective mapping f from the vertices of G to the vertices of H . The dilation of the embedding is the maximum distance between f(u),f(v) taken over all edges (u,v) of G .

We give a construction of embeddings of dilation 1 of complete binary trees into star graphs. The height of the trees embedded with dilation 1 into the n -dimensional star graph is Ω (n log n) , which is asymptotically optimal. Constructions of embeddings of complete binary trees of dilation 2δ and 2δ +1 , δ≥ 1, into star graphs are given. The use of larger dilation allows embeddings of trees of greater height into star graphs. It is shown that all these constructions can be modified to yield embeddings of dilation 1 and 2δ , δ≥ 1 , of complete binary trees into pancake graphs.