Decremental SPQR-trees for Planar Graphs

06/28/2018

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by   Jacob Holm, et al.

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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 Ω(n) operations, is O(^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(^2 n) amortized time. This is an exponential improvement over the previous best bound of O(√(n) ) that has stood for over 20 years. In addition, the previous data structures only supported edge deletions.