Graph Minors and Minimum Degree

Details Graph Minors and Minimum Degree Graph Minors and Minimum Degree

2008-12-05

arxiv.org/abs/0812.1064v1

Electronic J. Combinatorics R151, 2010

Mathematics

Combinatorics

Scientific paper

Let $\mathcal{D}_k$ be the class of graphs for which every minor has minimum degree at most $k$. Then $\mathcal{D}_k$ is closed under taking minors. By the Robertson-Seymour graph minor theorem, $\mathcal{D}_k$ is characterised by a finite family of minor-minimal forbidden graphs, which we denote by $\widehat{\mathcal{D}}_k$. This paper discusses $\widehat{\mathcal{D}}_k$ and related topics. We obtain four main results: We prove that every $(k+1)$-regular graph with less than ${4/3}(k+2)$ vertices is in $\widehat{\mathcal{D}}_k$, and this bound is best possible. We characterise the graphs in $\widehat{\mathcal{D}}_{k+1}$ that can be obtained from a graph in $\widehat{\mathcal{D}}_k$ by adding one new vertex. For $k\leq 3$ every graph in $\widehat{\mathcal{D}}_k$ is $(k+1)$-connected, but for large $k$, we exhibit graphs in $\widehat{\mathcal{D}}_k$ with connectivity 1. In fact, we construct graphs in $\mathcal{D}_k$ with arbitrary block structure. We characterise the complete multipartite graphs in $\widehat{\mathcal{D}}_k$, and prove analogous characterisations with minimum degree replaced by connectivity, treewidth, or pathwidth.

Affiliated with

Also associated with

No associations

Rating

Graph Minors and Minimum Degree does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Graph Minors and Minimum Degree, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Graph Minors and Minimum Degree will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-364985