Three-Terminal Reachability-Preserving Minimum Node Cut: Planar Hardness and a General-Graph \(O(\sqrt n)\)-Approximation

2026-06-12

AI Summary

This study addresses the three-terminal connectivity-preserving minimum node cut problem: removing non-terminal nodes of minimum total weight to disconnect a target terminal from two protected terminals, while preserving the connectivity between the protected pair. This problem arises in applications such as biological intervention, image analysis, and cybersecurity. The authors establish for the first time that the weighted version on planar graphs is NP-complete, thereby settling its computational complexity. They also present the first O(√n)-approximation algorithm for general graphs, which leverages an exact path-cut duality, a directed split-graph representation, and a submodular cut-function approximation technique achieving a square-root linear bound. The algorithm runs in polynomial time and matches the established hardness threshold for planar instances.

Abstract

We study the three-terminal reachability-preserving minimum node cut problem (\RPMNC). The input is an undirected graph \(G=(V,E)\), nonnegative vertex weights on nonterminal vertices, two protected terminals \(s_1,s_2\), and a target terminal \(t\). The goal is to delete a minimum-weight set of nonterminal vertices so that \(t\) is disconnected from the protected terminals, while \(s_1\) and \(s_2\) remain connected. This problem captures a basic ``separate while preserve'' requirement that arises in biological intervention design, image analysis with connectivity constraints, and cyber-security attack graph mitigation, where deleting or blocking a node represents preventing the corresponding action, state, or biological entity from participating in a harmful pathway. We prove two results. First, the weighted planar version of three-terminal \RPMNC{} is NP-complete. The reduction is from \textsc{Independent Set} on 3-regular Hamiltonian planar graphs and uses a one-sided blocker construction. Second, we give a polynomial-time \(O(\sqrt n)\)-approximation algorithm for general graphs. The algorithm is based on an exact path--separator identity, a directed split-graph representation of rooted vertex separators, and a root-linear approximation of a monotone submodular separator function.

Problem

• reachability-preserving
• minimum node cut
• three-terminal
• vertex separator
• graph connectivity

Innovation

• reachability-preserving
• minimum node cut
• planar hardness
• submodular separator
• approximation algorithm