Fixed Hypercube Embedding

In this paper, we consider the problem of embedding a given graph into a fixed-size hypercube. This work is related to other recent research on hypercube embedding. In [1], Bhat describes an order IVl log2 IVI algorithm for testing whether a given graph G = (V, E) is exactly an n-cube. Deciding whether a graph has a distance-preserving embedding into a hypercube can be done in polynomial time [3,7]. On the other hand, the present authors have shown that the problem of deciding whether a given graph is embeddable into any dimensioned hypercube is NP-complete [5]. Here we are concerned with the important practical problem of deciding whether a graph can be embedded into a fixed-size hypercube, thereby in a sense filling the gap between Bhat's result and that in [4]. We show that this problem is NP-complete. Our reduction uses the well-known 3-partition problem [4], which incidently was used in [6] to