A cactus graph is a graph in which any two cycles are edge-disjoint. We present a constructive proof of the fact that any plane graph G contains a cactus subgraph C where C contains at least a 1/6 fraction of the triangular faces of G. We also show that this ratio cannot be improved by showing a tight lower bound. Together with an algorithm for linear matroid parity, our bound implies two approximation algorithms for computing "dense planar structures" inside any graph: (i) A 1/6 approximation algorithm for, given any graph G, finding a planar subgraph with a maximum number of triangular faces; this improves upon the previous 1/11-approximation; (ii) An alternate (and arguably more illustrative) proof of the 4/9 approximation algorithm for finding a planar subgraph with a maximum number of edges. Our bound is obtained by analyzing a natural local search strategy and heavily exploiting the exchange arguments. Therefore, this suggests the power of local search in handling problems of this kind.
@InProceedings{chalermsook_et_al:LIPIcs.STACS.2019.19, author = {Chalermsook, Parinya and Schmid, Andreas and Uniyal, Sumedha}, title = {{A Tight Extremal Bound on the Lov\'{a}sz Cactus Number in Planar Graphs}}, booktitle = {36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-100-9}, ISSN = {1868-8969}, year = {2019}, volume = {126}, editor = {Niedermeier, Rolf and Paul, Christophe}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.19}, URN = {urn:nbn:de:0030-drops-102583}, doi = {10.4230/LIPIcs.STACS.2019.19}, annote = {Keywords: Graph Drawing, Matroid Matching, Maximum Planar Subgraph, Local Search Algorithms} }
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