Decremental SPQR-trees for Planar Graphs

TL;DR

This paper introduces a decremental data structure for planar graphs that efficiently maintains SPQR-trees and 3-vertex connectivity, significantly improving update times over previous methods.

Contribution

It presents the first decremental data structure for SPQR-trees in planar graphs with polylogarithmic update time, enabling fast connectivity queries.

Findings

Update time for SPQR-trees is $O( log^2 n)$ per operation.

Supports $O(1)$ query time for 3-vertex connectivity.

Achieves exponential improvement over previous $O(\sqrt{n})$ bounds.

Abstract

We present a decremental data structure for maintaining the SPQR-tree of a planar graph subject to edge contractions and deletions. The update time, amortized over $\Omega(n)$ operations, is $O(\log^2 n)$. Via SPQR-trees, we give a decremental data structure for maintaining $3$-vertex connectivity in planar graphs. It answers queries in $O(1)$ time and processes edge deletions and contractions in $O(\log^2 n)$ amortized time. This is an exponential improvement over the previous best bound of $O(\sqrt{n}\,)$ that has stood for over 20 years. In addition, the previous data structures only supported edge deletions.