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Decremental SPQR-trees for Planar Graphs

Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Lacki, Eva Rotenberg, Michael Wagner
2018 European Symposium on Algorithms  
Via SPQR-trees, we give a decremental data structure for maintaining 3-vertex connectivity in planar graphs.  ...  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(log 2 n).  ...  E S A 2 0 1 8 46:12 Decremental SPQR-trees for Planar Graphs Let us now analyze the running time.  ... 
doi:10.4230/lipics.esa.2018.46 dblp:conf/esa/HolmIKLR18 fatcat:styzxed7dffvzlf5lp4zem25uu

Decremental SPQR-trees for Planar Graphs [article]

Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Łącki, Eva Rotenberg
2018 arXiv   pre-print
Via SPQR-trees, we give a decremental data structure for maintaining 3-vertex connectivity in planar graphs.  ...  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).  ...  The useful concept of a separation is well-defined, even for general graphs: XX:6 Decremental SPQR-trees for Planar Graphs G = G[V ], G = G[V ] cover G, and V \ V and V \ V are both nonempty.  ... 
arXiv:1806.10772v1 fatcat:fu4y76zo2faj3dxqmdw2gfyreu

Incremental Convex Planarity Testing

Giuseppe Di Battista, Roberto Tamassia, Luca Vismara
2001 Information and Computation  
We present a data structure for the on-line incremental convex planarity testing problem with the following performance, where n denotes the current number of vertices of the graph: (strictly) convex planarity  ...  A planar graph is said to be convex planar if it admits a convex drawing.  ...  ACKNOWLEDGMENTS We thank the anonymous referees for their helpful comments and suggestions on how to improve the presentation.  ... 
doi:10.1006/inco.2001.3031 fatcat:psh2pkrsgzfxjg2wdy2b5mgmze

Front Matter, Table of Contents, Preface, Conference Organization

Yossi Azar, Hannah Bast, Grzegorz Herman, Michael Wagner
2018 European Symposium on Algorithms  
The best paper award for Track A was given to Jacob Holm, Giuseppe F. Italiano, Adam Karczmarz, Jakub Łącki and Eva Rotenberg for the paper "Decremental SPQR-trees for Planar Graphs".  ...  Jiang . . . . . . . . . . . . . . . . . . . . 15:1-15:13 Near-Optimal Distance Emulator for Planar Graphs Hsien-Chih Chang, Paweł Gawrychowski, Shay Mozes, and Oren Weimann . . . . . . 16:1-16:17 Approximation  ...  In response to the call for papers for ESA 2018, 307 papers were submitted, 256 for Track A and 51 for Track B (these are the counts after the removal of papers with invalid format and after withdrawals  ... 
doi:10.4230/lipics.esa.2018.0 dblp:conf/esa/X18 fatcat:jwouq3iofvhpvfxddhdsdgaip4

Good r-divisions Imply Optimal Amortised Decremental Biconnectivity [article]

Jacob Holm, Eva Rotenberg
2019 arXiv   pre-print
As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs  ...  We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n) time and handles any series of edge-deletions in O(  ...  , was used to obtain amortized O(1) decremental connectivity for planar graphs.  ... 
arXiv:1808.02568v2 fatcat:cgagyqobrbfvdjzj42potvcdnm

Simple Dynamic Spanners with Near-optimal Recourse against an Adaptive Adversary [article]

Sayan Bhattacharya, Thatchaphol Saranurak, Pattara Sukprasert
2022 arXiv   pre-print
Designing dynamic algorithms against an adaptive adversary whose performance match the ones assuming an oblivious adversary is a major research program in the field of dynamic graph algorithms.  ...  More precisely, for any k≥1, our algorithm maintains a (2k-1)-spanner of size O(n^1+1/klog n) with O(log n) amortized recourse, which is optimal in all parameters up to a O(log n) factor.  ...  STOC, pages 167-180.ACM, 2020. 2 59 Jacob Holm and Eva Rotenberg.Worst-case polylog incremental spqr-trees: Embeddings, planarity, and triconnectivity.In Proceedings of the 2020 ACM-SIAM Symposium on  ... 
arXiv:2207.04954v1 fatcat:5xlrgo3yizg55gglswz2k2h24a

Good r-Divisions Imply Optimal Amortized Decremental Biconnectivity

Jacob Holm, Eva Rotenberg, Markus Bläser, Benjamin Monmege
2021
As an immediate consequence, this gives optimal amortized decremental biconnectivity, 2-edge connectivity, and connectivity for large classes of graphs, including planar graphs and other minor free graphs  ...  We present a data structure that, given a graph G of n vertices and m edges, and a suitable pair of nested r-divisions of G, preprocesses G in O(m+n) time and handles any series of edge-deletions in O(  ...  Acknowledgements We are thankful to Adam Karczmarz and Jakub Łącki for their encouragement and interest in this work.  ... 
doi:10.4230/lipics.stacs.2021.42 fatcat:t3alkq27p5hktp45egw75gl4qq

Triconnected Planar Graphs of Maximum Degree Five are Subhamiltonian

Michael Hoffmann, Boris Klemz, Universitätsbibliothek Der FU Berlin
2020
The proof is constructive and yields a quadratic time algorithm to obtain a subhamiltonian plane cycle for a given graph.  ...  We show that every triconnected planar graph of maximum degree five is subhamiltonian planar.  ...  Our algorithm uses a decomposition of the graph into its triconnected components, which can be efficiently maintained via the SPQR-tree data structure [13, 15] .  ... 
doi:10.17169/refubium-28527 fatcat:xhjjv6ba7rc77gehkkfoqr2g5y

Triconnected Planar Graphs of Maximum Degree Five are Subhamiltonian

Michael Hoffmann, Boris Klemz, Michael A. Bender, Ola Svensson, Grzegorz Herman
2019
The proof is constructive and yields a quadratic time algorithm to obtain a subhamiltonian plane cycle for a given graph.  ...  We show that every triconnected planar graph of maximum degree five is subhamiltonian planar.  ...  Our algorithm uses a decomposition of the graph into its triconnected components, which can be efficiently maintained via the SPQR-tree data structure [13, 15] .  ... 
doi:10.3929/ethz-b-000378782 fatcat:m2zsg6ouzbep7mgd72w6o4qtzy