The algebra of metric betweenness I: Subdirect representation and retraction

We bring together algebraic concepts such as equational class and various concepts from graph theory for developing a structure theory for graphs that promotes a deeper analysis of their metric properties. The basic operators are Cartesian multiplication and gated amalgamation or, alternatively, retraction. Specifically, finite weakly median graphs are known to be decomposable (relative to these operators) into smaller pieces that in turn are parts of hyperoctahedra, the pentagonal pyramid, or of certain triangulations of the plane. This decomposition scheme can be interpreted as Birkhoff’s subdirect representation in purely algebraic terms.