# Bundled Crossings Revisited

## TL;DR

This paper investigates the problem of minimizing bundled crossings in graph drawings, proving NP-hardness for simple drawings and providing a fixed-parameter tractable algorithm for circular layouts, leveraging the connection to graph genus.

## Contribution

It establishes NP-hardness of bundled crossing minimization for simple drawings and introduces an FPT algorithm for circular layouts based on graph genus.

## Key findings

- NP-hardness of bundled crossing minimization in simple drawings
- FPT algorithm for circular layouts based on parameter k
- Connection between bundled crossings and graph genus

## Abstract

An effective way to reduce clutter in a graph drawing that has (many) crossings is to group edges that travel in parallel into \emph{bundles}. Each edge can participate in many such bundles. Any crossing in this bundled graph occurs between two bundles, i.e., as a \emph{bundled crossing}. We consider the problem of bundled crossing minimization: A graph is given and the goal is to find a bundled drawing with at most $k$ bundled crossings. We show that the problem is NP-hard when we require a simple drawing. Our main result is an FPT algorithm (in $k$) when we require a simple circular layout. These results make use of the connection between bundled crossings and graph genus.