In the following, a graph resulting from a join operation specified by the ILP described in the main text will be referred to as ILP-join-graph. The first two constraints only allow one connection per row and column, which is infringed in the first example by e 3 and f 3 , which both give rise to two newly formed edges. The third constraint aligns the join to impede multigraph formation, as it is depicted in the second example. The fourth constraint ensures that the edge labels of introduced edges adhere to the definition of a weight preserving join, violated in the forth example.