Interconnection networks play an important role in parallel computing systems. A lot of effort has been made to construct rearrangeable networks. This paper generalizes the concept of rearrangeability to general, directed graphs. Given a directed graph G =(V,E ), where V = 1, 2, ..., n, an n-permutation π is said to be realizable on G if there exist n edge-disjoint paths in G that connect vertex i to vertex π(i) (1 ≤ i ≤ n). Graph G is said to be rearrangeable if all the n!n-permutations are realizable on G. This paper presents tight lower bounds on the number of edges in a rearrangeable graph. One of the lower bounds is m ≥ 2(n-1), where m and n are the number of edges and the number of vertices, respectively. Moreover, this paper introduces a new graph parameter, rearrangeable number, denoted by ψ, which is the minimal multiplicity every edge in G needs to be duplicated so that the resultant graph becomes rearrangeable. A lower bound on the value of ψ is derived in the paper.
Key wordsPermutation capability Permutation realization Rearrangeable graph
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