# An exponential lower bound for cut sparsifiers in planar graphs

## TL;DR

This paper proves that in planar graphs, any cut mimicking network must sometimes be exponentially large, highlighting fundamental limits in graph compression for certain graph classes.

## Contribution

It establishes an exponential lower bound for cut mimicking networks in planar graphs, nearly matching known upper bounds and contrasting with simpler cases.

## Key findings

- Existence of planar graphs requiring exponential edges in mimicking networks
- Nearly tight lower bounds for cut mimicking network size in planar graphs
- Identification of hard instances for existing upper bound constructions

## Abstract

Given an edge-weighted graph $G$ with a set $Q$ of $k$ terminals, a mimicking network is a graph with the same set of terminals that exactly preserves the sizes of minimum cuts between any partition of the terminals. A natural question in the area of graph compression is to provide as small mimicking networks as possible for input graph $G$ being either an arbitrary graph or coming from a specific graph class. In this note we show an exponential lower bound for cut mimicking networks in planar graphs: there are edge-weighted planar graphs with $k$ terminals that require $2^{k-2}$ edges in any mimicking network. This nearly matches an upper bound of $O(k 2^{2k})$ of Krauthgamer and Rika [SODA 2013, arXiv:1702.05951] and is in sharp contrast with the $O(k^2)$ upper bound under the assumption that all terminals lie on a single face [Goranci, Henzinger, Peng, arXiv:1702.01136]. As a side result we show a hard instance for the double-exponential upper bounds given by Hagerup, Katajainen, Nishimura, and Ragde~[JCSS 1998], Khan and Raghavendra~[IPL 2014], and Chambers and Eppstein~[JGAA 2013].