Edge-splittings preserving local edge-connectivity of graphs

Abstract

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.