A Tight Extremal Bound on the Lov\'{a}sz Cactus Number in Planar Graphs

TL;DR

This paper proves a tight lower bound on the fraction of triangular faces in planar graphs that can be covered by a cactus subgraph, and introduces improved approximation algorithms for dense planar substructures.

Contribution

It provides a constructive proof of a tight bound on the Lovász cactus number in planar graphs and develops improved approximation algorithms for dense planar subgraph problems.

Findings

A  approximation for maximum triangular face planar subgraphs.

A /9 approximation for maximum edge planar subgraphs.

Demonstrates the effectiveness of local search strategies.

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.