A Spanning-Tree-Based Algorithm for Planar Graph Dismantling

TL;DR

This work tackles edge-budget planar graph dismantling by introducing a spanning-tree skeleton that captures the network's connectivity backbone. It develops a dual-path, budget-aware framework that chooses between a density-informed small-budget path and a slope-prior large-budget path, guided by a baseline bisection on a Uniform Spanning Tree and a density feature $t=\log|E|/\log|V|$. The subproblem solution relies on sampling $m$ Uniform Spanning Trees via Wilson’s algorithm and a one-time post-order greedy partition, yielding a scalable $O(|E|)$ per-trial method that aggregates into a near-linear overall complexity $O(m|E||K|)$. Empirical results on random planar graphs demonstrate efficient fragmentation with favorable LCC reductions and strong parallelism, supporting practical applicability to spatial robustness analysis and network interdiction planning.

Abstract

In spatially embedded networks such as transportation and power grids, understanding how edge removals affect connectivity is crucial for robustness analysis. This paper studies a planar graph dismantling problem under an edge-budget constraint. We propose a spanning-tree-skeleton dual-path framework that first samples multiple uniform spanning trees to capture network backbones and then adaptively selects between two complementary paths according to the budget. The small-budget path estimates a dismantlable subgraph fraction using a logarithmic density feature, while the large-budget path predicts the optimal partition count through a slope-based model. Experiments on random planar graphs demonstrate near-linear runtime scaling, consistent reductions in the largest connected component ratio, and clear budget-fragmentation trends. The method provides an interpretable and efficient approach for planar-network robustness analysis.

Figures (6)

• Figure 1: Illustration of planar graph bisection based on a spanning tree skeleton. (a) The original graph $G(V,E)$ with 200 nodes and 550 edges. (b) A spanning tree skeleton sampled from the original graph. (c) The balanced bisection result. The spanning-tree-based planar bisection yields a baseline cut cost $|F^{(2)}|$, which, together with the given budget $cut$, determines the subsequent algorithm path.
• Figure 2: Example of the small-budget path process. (a) A spanning tree sampled under the estimated fraction $\hat{\alpha}$. (b) The subgraph induced by the spanning tree skeleton. (c) The bipartition result on the subgraph. (d) The corresponding partitioning result mapped onto the original graph. This process involves only a single spanning tree and local subgraph operations, yielding a fragmentation outcome quickly with complexity on the order of $O(|E|)$.
• Figure 3: Partition examples on a small-scale planar graph under different target partition counts. As $k$ increases, the graph is split into more balanced, smaller components, and the LCC decreases.
• Figure 4: Small-scale dataset results – joint trends of budget ratio ($cut/|E|$, left axis) and LCC size ratio ($L_{\max}/|V|$, right axis) as $k$ increases. Curves represent means across random seeds; shaded areas show standard deviation.
• Figure 5: Partition examples on a large-scale planar graph for different $k$. As $k$ increases, the number of partitions grows and LCC shrinks, with balanced fragmentation across components.