Efficient Embeddings into Hypercube-like Topologies

Embeddings of various graph classes into hypercubes have been widely studied. Almost all these classes are regularly structured graphs such as meshes, complete trees or pyramids. In this paper, we present a general method for one-to-one embeddings of irregularly structured graphs into their optimal hypercubes, based on extended edge bisectors of graphs. An extended edge bisector is an edge bisector with the additional property that a certain subset of the vertices is distributed more or less evenly among the two halves of the bisected graph. The dilation and congestion of our embedding depends on the quality of the extended edge bisector. Moreover, if the extended bisection can be computed efficiently on the hypercube, then so can the embedding. Our embedding technique can also be applied to embeddings into hypercube-like topologies such as folded hypercubes, twisted cubes, crossed cubes, Möbius cubes, Fibonacci cubes or star graphs.