  Let $\lambda(G)$ be 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.
Let $\lambda_{\rm e}(G)$ be the smallest number of edges that can be removed
from $G$ for the same purpose. Let $k$ be the maximum degree of $G$, let $t$ be
the number of vertices of degree $k$, let $M(G)$ be the set of vertices of
degree $k$, let $n$ be the number of vertices in the closed neighbourhood of
$M(G)$, and let $m$ be the number of edges incident to vertices in $M(G)$.
Fenech and the author showed that $\lambda(G) \leq \frac{n+(k-1)t}{2k}$, and
they essentially showed that $\lambda (G) \leq n \left ( 1- \frac{k}{k+1} {
\Big( \frac{n}{(k+1)t} \Big) }^{1/k} \right )$. They also showed that
$\lambda_{\rm e}(G) \leq \frac{m + (k-1)t}{2k-1}$ and $\lambda_{\rm e} (G) \leq
m \left ( 1- \frac{k-1}{k} { \Big( \frac{m}{kt} \Big) }^{1/(k-1)} \right )$.
These bounds are attained if $k \geq 2$ and $G$ is the union of $t$ pairwise
vertex-disjoint $(k+1)$-vertex stars. For each of $\lambda(G)$ and
$\lambda_{\rm e}(G)$, the two bounds on the parameter are compared for the
purpose of determining, for each bound, the cases in which the bound is better
than the other. This work is also motivated by the likelihood that similar
pairs of bounds will be discovered for other graph parameters and the same
analysis can be applied.
