Bombing of Königsberg problem

Color the vertices with an odd number of edges white, and the other vertices black. We consider sets of paths, each path from a white vertex to another white vertex. Such a set of paths with a minimal number of edges, that still connects to every white vertex, constitutes a minimum number of bridges to destroy that leaves an Eulerian cycle.

If instead we consider such a set of paths with a minimal number of edges, that connects to every white vertex but two, that will answer the original question, giving the minimal number of bridges to destroy that leaves an Eulerian path.