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#### Abstract

Let $\lambda(G)$ denote the smallest number of vertices that can be removed from a non-empty graph $G$ so that the resulting graph has a smaller maximum degree. In a recent paper, we proved that if $n$ is the number of vertices of $G$, $k$ is the maximum degree of $G$, and $t$ is the number of vertices of degree $k$, then $\lambda (G) \leq \frac{n+(k-1)t}{2k}$. We also showed that $\lambda (G) \leq \frac{n}{k+1}$ if $G$ is a tree. In this paper, we provide a new proof of the first bound and use it to determine the graphs that attain the bound, and we also determine the trees that attain the second bound.

#### Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

#### Recommended Citation

Borg, Peter and Fenech, Kurt
(2018)
"Reducing the maximum degree of a graph by deleting vertices: the extremal cases,"
*Theory and Applications of Graphs*: Vol. 5
:
Iss.
2
, Article 5.

DOI: 10.20429/tag.2018.050205

Available at:
https://digitalcommons.georgiasouthern.edu/tag/vol5/iss2/5

*Supplemental file with DOI*