Augmenting graphs to meet edge-connectivity requirements
SIAM Journal on Discrete Mathematics
Discrete Mathematics - Special volume (part two) to mark the centennial of Julius Petersen's “Die theorie der regula¨ren graphs” (“The theory of regular graphs”)
Edge-Connectivity Augmentation Preserving Simplicity
SIAM Journal on Discrete Mathematics
Edge-Connectivity Augmentation with Partition Constraints
SIAM Journal on Discrete Mathematics
Journal of Combinatorial Theory Series B
Efficient edge splitting-off algorithms maintaining all-pairs edge-connectivities
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
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Let G=(V+s,E) be a 2-edge-connected graph with a designated vertex s. A pair of edges rs,st is called admissible if splitting off these edges (replacing rs and st by rt) preserves the local edge-connectivity (the maximum number of pairwise edge disjoint paths) between each pair of vertices in V. The operation splitting off is very useful in graph theory, it is especially powerful in the solution of edge-connectivity augmentation problems as it was shown by Frank [Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Math. 5(1) (1992) 22-53]. Mader [A reduction method for edge-connectivity in graphs, Ann. Discrete Math. 3 (1978) 145-164] proved that if d(s)3 then there exists an admissible pair incident to s. We generalize this result by showing that if d(s)=4 then there exists an edge incident to s that belongs to at least @?d(s)/3@? admissible pairs. An infinite family of graphs shows that this bound is best possible. We also refine a result of Frank [On a theorem of Mader, Discrete Math. 101 (1992) 49-57] by describing the structure of the graph if an edge incident to s belongs to no admissible pairs. This provides a new proof for Mader's theorem.