Chapter 20. Isometrie Embeddings of Graphs into Cartesian Produets
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Chapter 20. Isometrie Embeddings of Graphs into Cartesian Produets
ametrie representation of G. Two isometrie embeddings of G into Cartesian .... edges e, fEE in relation by fJ, let T be a spanning tree eontaining both e and.
Chapter 20. Isometrie Embeddings of Graphs into Cartesian Produets
We have characterized in the previous chapter the graphs that can be isometrically embedded into a hypercube. The hypercube is the simplest example of a Cartesian product of graphsj indeed, the m-hypercube is nothing but (K2 )m. We consider here isometrie embeddings of graphs into arbitrary Cartesian products. It turns out that every graph can be isometrically embedded in a canonical way into a Cartesian product whose factors are "irreducible", i.e., eannot be further embedded into Cartesian products. We present two applications of this result, for finding the prime factorization of a graph, and for showing that the path metrie of every bipartite graph can decomposed in a unique way as a nonnegative eombination of primitive semimetrics.
20.1
Canonical Metric Representation of a Graph
Let G, H b ... ,Hk be graphs. An isometrie embedding of G into the Cartesian product TIl
