• Corpus ID: 212634159

# 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}
}
• Published 9 March 2020
• Mathematics
• arXiv: Combinatorics
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
• Mathematics
Electron. Notes Discret. Math.
• 2009
Trees and Co-trees with Bounded Degrees in Planar 3-connected Graphs
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.
2-connected Coverings of Bounded Degree in 3-connected Graphs
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
• Mathematics
J. 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
• Mathematics
Graphs 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
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
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
Hamilton paths in toroidal graphs
• Mathematics
J. Comb. Theory, Ser. B
• 2005