Detachment of vertices of graphs preserving edge-connectivity

Balázs Fleiner

Published in:
SIAM J. Discrete Math. 18 (2004) 581-591


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 resulted graph is k-edge-connected.

Bibtex entry:

AUTHOR = {Fleiner, Bal{\'a}zs},
TITLE = {Detachment of vertices of graphs preserving edge-connectivity},
NOTE= {{\tt}},
INSTITUTION = {Egerv{\'a}ry Research Group, Budapest},
YEAR = {2002},
NUMBER = {TR-2002-01}

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