Finally, we need to ensure that we do not introduce new multi-edges into the join graph K. For a vertex u ∈ U ′ , let E u annotate the edges of u that are incident to u in C G (analogously for w ∈ W ′ and E w ). Since the introduction of multi-edges to K can only occur for vertices where E u and E w , respectively, contain more than one edge, we obtain the following constraint: