The authors consider the problem of constructing a minimum-weight, two-connected network spanning all the points in a set V. They assume a symmetric, nonnegative distance function d(ë) defined on V×V which satisfies the triangle inequality. The authors obtain a structural characterization of optimal solutions. Specifically, there exists an optimal two-connected solution whose vertices all have degree 2 or 3, and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components. These are the strongest possible conditions on the structure of an optimal solution since the authors also show that any two-connected graph satisfying these conditions is the unique optimal solution for a particular choice of ‘canonical’ distances satisfying the triangle inequality. They use these properties to show that the weight of an optimal traveling salesman cycle is at most 4/3 times the weight of an optimal two-connected solution; examples are provided which approach this bound arbitrarily closely. In addition, the authors obtain similar results for the variation of this problem where the network need only span a prespecified subset of the points.
