Cut Tree Structures with Applications on Contraction-Based Sparsification

We introduce three new cut tree structures of graphs G in which the vertex set of the tree is a partition of V (G) and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently, Kawarabayashi and Thorup [8] presented the first deterministic near-linear edge-connectivity recognition algorithm. A crucial step in this algorithm uses the existence of vertex subsets of a simple graph G whose contractions leave a graph with Õ(n/δ) vertices and Õ(n) edges (n := |V (G)|) such that all non-trivial min-cuts of G are preserved. We improve this result by eliminating the poly-logarithmic factors, that is, we show a contraction-based sparsification that leaves O(n/δ) vertices and O(n) edges and preserves all non-trivial min-cuts. We complement this result by giving a sparsification that leaves O(n/δ) vertices and O(n) edges such that all (possibly not minimum) cuts of size less than δ are preserved, by using contractions in a second tree structure. As consequence, we have that every simple graph has O(n/δ) δ-edge-connected components, and, if it is connected, it has O((n/δ)2) non-trivial min-cuts. All these results are proven to be asymptotically optimal. By using a third tree structure, we give a new lower bound on the number of pendant pairs (that is, pairs of vertices v, w with λ(v, w) = min{d(v), d(w)}). The previous best bound was given 1974 by Mader, who showed that every simple graph contains Ω(δ2) pendant pairs. We improve this result by showing that every simple graph G with δ ≥ 5 or λ ≥ 4 or κ ≥ 3 contains Ω(δn) pendant pairs. We prove that this bound is asymptotically tight from several perspectives, and that Ω(δn) pendant pairs can be computed efficiently.