On minimally k-edge-connected graphs and shortest k-edge-connected Steiner networks

Authors:
Tibor Jordán
Affiliations:
Department of Operations Research, Eötvös University, Pázmány Péter sétány 1/C, Budapest, H-1117, Hungary
Venue:
Discrete Applied Mathematics - Submodularity
Year:
2003

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Abstract

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.