Before revision:  Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle . That is, we prove that for any decomposition D that contains at least one odd cycle, there exists a motif H_{D . These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.
After revision:  Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle . These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition.