Compacting Graphs According to Adjacencies in Linear Time and Space

Given an (un)directed connected graph G = (V; E) as a set of edges, let ?(v) be the set of neighbors of a vertex v 2 V. We show that the equivalence-relation R with uRv :, ?(u) = ?(v) on the vertices of the graph can be computed in linear time and space. Many problems can be solved on the graph G induced by the equivalence-classes and the result extended to G. An example for this is the all-pair-shortest-path problem. Another application can be found in the eld of parallel simulation of neural nets.