# Rooted Minors and Locally Spanning Subgraphs

@article{Bohme2020RootedMA, title={Rooted Minors and Locally Spanning Subgraphs}, author={Thomas Bohme and Jochen Harant and Matthias Kriesell and Samuel Mohr and Jens M. Schmidt}, journal={arXiv: Combinatorics}, year={2020} }

Given a graph $G$ and $X\subseteq V(G)$, we say $M$ is an minor of $G$ rooted at $X$, if $M$ is a minor of $G$ such that each bag contains at most one vertex of $X$ and $X$ is a subset of the union of all bags. We consider the problem whether $G$ has a highly connected minor rooted at $X$ if $X\subseteq V(G)$ cannot be separated in $G$ by removing a few vertices of $G$.
Our results constitute a general machinery for strengthening statements about $k$-connected graphs (for $1 \leq k \leq 4$) to…

## References

SHOWING 1-10 OF 25 REFERENCES

Spanning trees in 3-connected K3, t-minor-free graphs

- MathematicsElectron. Notes Discret. Math.
- 2009

Trees and Co-trees with Bounded Degrees in Planar 3-connected Graphs

- MathematicsSWAT
- 2014

It is shown that every planar 3-connected graph has a spanning tree T such that both T and its co-tree have maximum degree at most 5.

Hamiltonian cycles through prescribed edges of 4-connected maximal planar graphs

- MathematicsDiscret. Math.
- 2010

2-connected Coverings of Bounded Degree in 3-connected Graphs

- MathematicsJ. Graph Theory
- 1995

It is shown that every 3-connected graph which is embeddable in the sphere, the projective plane, the torus or the Klein bottle has a 2-connected spanning subgraph of maximum degree at most six.

Menger's Theorem

- Mathematics
- 2001

Menger's Theorem for digraphs states that for any two vertex sets A and B of a digraph D such that A cannot be separated from B by a set of at most t vertices, there are t + 1 disjoint A–B-paths in…

Menger's Theorem

- MathematicsJ. Graph Theory
- 2001

A short and elementary proof of a more general theorem is given of Menger's Theorem for digraphs, which states that for any two vertex sets A and B of a digraph D such that A cannot be separated from B by a set of at most t vertices, there are t + 1 disjoint A-B- paths in D.

Spanning Trees: A Survey

- MathematicsGraphs Comb.
- 2011

This paper mainly deals with spanning trees having some particular properties concerning a hamiltonian properties, for example, spanning trees with bounded degree, with bounded number of leaves, or with boundedNumber of branch vertices.

2-Connected Spanning Subgraphs of Planar 3-Connected Graphs

- MathematicsJ. Comb. Theory, Ser. B
- 1994

Abstract We prove that every planar 3-connected graph has a 2-connected spanning subgraph of maximum valence 15. We give an example of a planar 3-connected graph with no spanning 2-connected subgraph…

On paths in planar graphs

- MathematicsJ. Graph Theory
- 1997

This paper generalizes a theorem of Thomassen on paths in planar graphs. As a corollary, it is shown that every 4-connected planar graph has a Hamilton path between any two specified vertices x, y…