Decomposition andl1-Embedding of Weakly Median Graphs

Weakly median graphs, being defined by interval conditions and forbidden induced subgraphs, generalize quasi-median graphs as well as pseudo-median graphs. It is shown that finite weakly median graphs can be decomposed with respect to gated amalgamation and Cartesian multiplication into 5-wheels, induced subgraphs of hyperoctahedra (alias cocktail party graphs), and 2-connected bridged graphs not containing K4or K1,1,3as an induced subgraph. As a consequence one obtains that every finite weakly median graph is l1-embeddable, that is, it embeds as a metric subspace into some Rnequipped with the 1-norm.