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Local consistency properties and algorithms for enforcing them are central to the success of Constraint Processing. Recently, we have demonstrated the importance of higher levels of consistency and the effectiveness of their algorithms for solving difficult problems (Karakashian et al. 2010; Woodward et al. 2011). In this paper, we introduce two reformulation techniques for improving the effectiveness of our algorithm for the relational consistency property R(*;m)C (Karakashian et al. 2010). Both techniques exploit a tree decomposition of the constraint network of a Constraint Satisfaction Problem (CSP), which is a tree embedding of the network. Our first reformulation technique exploits the structure of the decomposition tree and the state of the backtrack search to omit unnecessary steps from our algorithm and improve its performance. Our second contribution is new relational consistency property called T-R(*; m; z)C that is strictly stronger than R(*;m)C. This property is achieved by modifying the structure of the constraint network and adding new redundant constraints to the CSP at the intersection of two vertices of the tree decomposition (Rollon and Dechter 2010). We demonstrate the advantages of the proposed two reformulations for finding all the solutions of a CSP using the technique known as Backtracking with Tree Decomposition (BTD) (Jégou and Terrioux 2003).