# The Densest k Subgraph Problem in b-Outerplanar Graphs

## TL;DR

This paper presents efficient algorithms for finding the densest k subgraph in b-outerplanar graphs and discusses the limitations of existing approximation techniques in planar graphs.

## Contribution

It introduces exact algorithms with specific time complexities for b-outerplanar graphs and explores the challenges of applying Baker's PTAS to planar graphs.

## Key findings

- Exact $O(nk^2)$ algorithm for outerplanar graphs
- Extended $O(nk^2 8^b)$ algorithm for b-outerplanar graphs
- Baker's PTAS likely ineffective for planar graphs

## Abstract

We give an exact $O(nk^2)$ algorithm for finding the densest k subgraph in outerplanar graphs. We extend this to an exact $O(nk^2 8^b)$ algorithm for finding the densest k subgraph in b-outerplanar graphs. Finally, we hypothesize that Baker's PTAS technique will not work for the densest k subgraph problem in planar graphs.