# Obstructions for partitioning into forests and outerplanar graphs

## TL;DR

This paper characterizes when graphs can be partitioned into forests or outerplanar graphs with bounded obstructions, introducing new parameters called edge-brittleness and vertex-brittleness, and relates these to forbidden obstructions and edit distance.

## Contribution

It introduces and characterizes boundedness of edge-brittleness and vertex-brittleness for graph classes related to forests and outerplanar graphs using forbidden obstructions.

## Key findings

- Characterization of classes with bounded $ ext{C}$-edge-brittleness for forests and graphs without $K_4\setminus e$ minors.
- Characterization of classes with bounded $ ext{C}$-vertex-brittleness for forests and outerplanar graphs.
- Analysis of the relationship between these parameters and edit distance.

## Abstract

For a class $\mathcal C$ of graphs, we define $\mathcal C$-edge-brittleness of a graph $G$ as the minimum $\ell$ such that the vertex set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal C$ and there are $\ell$ edges having ends in distinct parts. We characterize classes of graphs having bounded $\mathcal C$-edge-brittleness for a class $\mathcal C$ of forests or a class $\mathcal C$ of graphs with no $K_4\setminus e$ topological minors in terms of forbidden obstructions. We also define $\mathcal C$-vertex-brittleness of a graph $G$ as the minimum $\ell$ such that the edge set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal C$ and there are $\ell$ vertices incident with edges in distinct parts. We characterize classes of graphs having bounded $\mathcal C$-vertex-brittleness for a class $\mathcal C$ of forests or a class $\mathcal C$ of outerplanar graphs in terms of forbidden obstructions. We also investigate the relations between the new parameters and the edit distance.