On the Complexity of Isometric Embedding in the Hypercube

A finite metric is h − embeddable if it can be embedded isometrically in the N -cube (hypercube) for some N . It is known that the problem of testing whether a metric is h − embeddable is NP-Complete, even if the distances are restricted to the set {2, 4, 6 }. Here we study the problem where the distances are restricted to the set {1, 2, 3 } and give a polynomial time algorithm and forbidden submetric characterisation. In fact, we show these metrics are h − embeddable if and only if they are 11 − gonal and the sum of the distances arround any triangle is even. The so-called truncated metric case, where the distances are chosen from {1, 2} is particularly simple, the only embeddable metrics arise from the graphs K1,n−1, K2,2, and 2Kn ( Kn with all distances 2).