Cartesian Products of Graphs and Metric Spaces

Abstract

We prove uniqueness of decomposition of a finite metric space into a product of metric spaces for a wide class of product operations. In particular, this gives the positive answer to the long-standing question of S. Ulam: 'If U x U ≃ V x V with U, V compact metric spaces, will then U and V be isometric?' in the case of finite metric spaces. In the proof we use uniqueness of cartesian decomposition of connected graphs; a known fact to which we give a new proof which is shorter and more transparent than existing ones. © 2000 Academic Press.