@online{Chalermsook_arXiv1804.03485,
TITLE = {A Tight Extremal Bound on the {Lov\'{a}sz} Cactus Number in Planar Graphs},
AUTHOR = {Chalermsook, Parinya and Schmid, Andreas and Uniyal, Sumedha},
LANGUAGE = {eng},
URL = {http://arxiv.org/abs/1804.03485},
EPRINT = {1804.03485},
EPRINTTYPE = {arXiv},
YEAR = {2018},
ABSTRACT = {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 $\frac{1}{6}$ 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 $\frac{1}{6}$ approximation algorithm for, given any graph $G$, finding a planar subgraph with a maximum number of triangular faces; this improves upon the previous $\frac{1}{11}$-approximation; (ii) An alternate (and arguably more illustrative) proof of the $\frac{4}{9}$ 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.},
}