# Distance Measures for Embedded Graphs

## TL;DR

This paper introduces new geometric distance measures for embedded graphs using Fréchet and weak Fréchet distances, providing algorithms for their computation and analyzing their computational complexity.

## Contribution

It defines novel graph distance measures based on continuous mappings and develops algorithms for their computation, including polynomial-time solutions for trees and planar graphs.

## Key findings

- Deciding these distances is NP-hard for general embedded graphs.
- Polynomial time algorithms exist for trees and planar embedded graphs with weak Fréchet distance.
- Approximation algorithms are provided for the NP-hard cases.

## Abstract

We introduce new distance measures for comparing straight-line embedded graphs based on the Fr\'echet distance and the weak Fr\'echet distance. These graph distances are defined using continuous mappings and thus take the combinatorial structure as well as the geometric embeddings of the graphs into account. We present a general algorithmic approach for computing these graph distances. Although we show that deciding the distances is NP-hard for general embedded graphs, we prove that our approach yields polynomial time algorithms if the graphs are trees, and for the distance based on the weak Fr\'echet distance if the graphs are planar embedded. Moreover, we prove that deciding the distances based on the Fr\'echet distance remains NP-hard for planar embedded graphs and show how our general algorithmic approach yields an exponential time algorithm and a polynomial time approximation algorithm for this case.