Published in:
SIAM J. Discrete Math. 18 (2004) 581-591
The detachment of vertex is the inverse operation of merging vertices s_{1},...,s_{t} into s. We speak about { d_{1},...,d_{t}}-detachment if for the detached graph G' the new degrees are specified as d_{G'}(s_{1})=d_{1},...,d_{G'}(s_{t})=d_{t}. We call a detachment k-feasible if d_{G'}(X) \geq 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 { d_{1},...,d_{t}}-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 \gamma 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.
Bibtex entry:
@techreport{egres-02-01,
AUTHOR | = | {Fleiner, Bal{\'a}zs}, |
TITLE | = | {Detachment of vertices of graphs preserving edge-connectivity}, |
NOTE | = | {{\tt www.cs.elte.hu/egres}}, |
INSTITUTION | = | {Egerv{\'a}ry Research Group, Budapest}, |
YEAR | = | {2002}, |
NUMBER | = | {TR-2002-01} |