Detachment of Vertices of Graphs Preserving Edge-Connectivity

The detachment of vertex is the inverse operation of merging vertices s1, . . . , st into s. We speak about {d1, . . . , dt}-detachment if for the detached graph G′ the new degrees are specified as dG′(s1) = d1, . . . , dG′(st) = dt. We call a detachment k-feasible if dG′(X) ≥ k whenever X separates two vertices of V (G)−s. In our main theorem, we give a necessary and sufficient condition for the existence of a k-feasible {d1, . . . , dt}-detachment of vertex s. This theorem also holds for graphs containing 3-vertex hyperedges disjoint from s. From special cases of the theorem, we get a characterization of those graphs whose edge-connectivity can be augmented to k by adding γ edges and p 3-vertex hyperedges. We give a new proof for the theorem of Nash-Williams that characterizes the existence of a simultaneous detachment of the vertices of a given graph such that the resulted graph is k-edge-connected.