There are various definitions of a gene cluster determined by two genomes and methods for finding these clusters. However, there is little work on characterizing configurations of genes that are eligible to be a cluster according to a given definition. For example, given a set of genes in a genome, is it always possible to find two genomes such that their intersection is exactly this cluster? In one version of this problem, we make use of the graph theory to reformulated it as follows: Given a

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... raph G with n vertices, do there exist two θ -powers of paths G S = (V S , E S ) and G T = (V T , E T ) such that G S ∩G T contains G as an induced subgraph? In this work, we divide the problem in two cases, depending on whether or not G is an induced subgraph of G S or G T . We show an O(n 2 ) time algorithm that generates the smallest θ -powers of paths G S and G T (with respect to and the number of vertices) that contains G as an induced subgraph. Finally, we discuss the problem when G is an induced subgraph neither of G S nor of G T and we present a method of finding the smallest power of a path when graph G is a cycle C n .