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)\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 {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 $\gamma$ 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 resulting graph is k-edge-connected.
[1]
Akira Nakamura,et al.
Edge-Connectivity Augmentation Problems
,
1987,
J. Comput. Syst. Sci..
[2]
András Frank.
Augmenting Graphs to Meet Edge-Connectivity Requirements
,
1992,
SIAM J. Discret. Math..
[3]
L. Lovász.
Combinatorial problems and exercises
,
1979
.
[4]
Tibor Jordán,et al.
Detachments Preserving Local Edge-Connectivity of Graphs
,
2003,
SIAM J. Discret. Math..
[5]
W. Mader.
A Reduction Method for Edge-Connectivity in Graphs
,
1978
.
[6]
C. Nash-Williams,et al.
Connected detachments of graphs and generalized Euler trails
,
1985
.