Abstract A binary CSP instance satisfying the brokentriangle property (BTP) can be solved in polynomial time. Unfortunately, in practice, few instances satisfy the BTP. We show that a local version of the BTP allows the merging of domain values in binary CSPs, thus providing a novel polynomial-time reduction operation. Experimental trials on benchmark instances demonstrate a significant decrease in instance size for certain classes of problems. We show that BTP-merging can be generalised to instances with constraints of arbitrary arity. A directional version of the general-arity BTP then allows us to extend the BTP tractable class previously defined only for binary CSP.