A Tight Extremal Bound on the Lov\'asz Cactus Number in Planar Graphs
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 16 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 16 approximation algorithm for, given any graph G, finding a planar subgraph with a maximum number of triangular faces; this improves upon the previous 111-approximation; (ii) An alternate (and arguably more illustrative) proof of the 49 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.