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 \cite{Kawarabayashi2015a} 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 $\tilde{O}(n/\delta)$ vertices and $\tilde{O}(n)$ 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 $O(n/\delta)$ vertices and $O(n)$ edges and preserves all non-trivial min-cuts. We complement this result by giving a sparsification that leaves $O(n/\delta)$ vertices and $O(n)$ edges such that all (possibly not minimum) cuts of size l 查看全文>>