The binary de Bruijn graph (BDG) is a realizable alternative to the hypercube. An extension of the BDG by the edge set union with a torus is shown-union-graph (UG). This achieves graph capabilities/versatility comparable to the product shuffle (PS) graph and the hypercube within a fixed degree graph. The UG improves upon both in implementing pipelined and multi-phase algorithms. More importantly, the purpose being to design an algorithmically specialized interconnect, with the algorithmic features of a wide range of algorithms as well as direct architectural support for them, instead of simple providing for a set of graph embeddings in the interconnect. A set of examples demonstrate the UG's versatility in this aspect of algorithmic support.