Data structure for node connectivity queries
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Let κ(s,t) denote the maximum number of internally disjoint paths in an undirected graph G. We consider designing a data structure that includes a list of cuts, and answers in O(1) time the following query: given s,t ∈ V, determine whether κ(s,t) ≤ k, and if so, return a pointer to an st-cut of size ≤ k in the list. A trivial data structure includes a list of n(n-1)/2 cuts and requires Θ(kn^2) space. We show that O(kn) cuts suffice, thus reducing the space to O(k^2 n+n^2). In the case when G is k-connected, we show that O(n) cuts suffice, and that these cuts can be partitioned into O(k) laminar families; this reduces the space to O(kn). The latter result slightly improves and substantially simplifies a recent result of Pettie and Yin [ICALP 2021].
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