We suspect that a natural join can be found with a non-trivial polynomial algorithm, although a proof remains as an open problem. The number of unique solutions is exponential in general, as it can be well demonstrated on instances where each cut node is connected to a respective single cut edge of the same weight. In the applications in later sections of this paper we are mainly interest in finding all (exponentially many) unique natural joins, which amounts to enumerating all possible recombinations of the parent fragments. We therefore opted for an implementation as an integer linear programming instance, which naturally lends itself for this purpose and yields a straightforward implementation. : Cut-and-join crossover of two planar graphs can be restricted to yield again a planar graph. Since every minimal cut of a planar graph is cycle in its complement, there is a circular order to the severed edges and thus of the vertices incident with the cutset. It sufficies to preserve the circular order of these vertices (in arbitrary orientation). The example here is in addition degree-preserving.