On minimally k-edge-connected graphs and shortest k-edge-connected Steiner networks Original Research Article

A graph G=(V,E) is called minimally (k,T)-edge-connected with respect to some T⊆V if there exist k-edge-disjoint paths between every pair u,v∈T but this property fails by deleting any edge of G. We show that |V| can be bounded by a (linear) function of k and |T| if each vertex in V−T has odd degree. We prove similar bounds in the case when G is simple and k⩽3. These results are applied to prove structural properties of optimal solutions of the shortest k-edge-connected Steiner network problem. We also prove lower bounds on the corresponding Steiner ratio.