A graph $G$ is degree-continuous if the degrees of every two adjacent vertices of $G$ differ by at most 1. A finite nonempty set $S$ of integers is convex if $k \in S$ for every integer $k$ with $\min (S) \le k \le \max (S)$. It is shown that for all integers $r > 0$ and $s \ge 0$ and a convex set $S$ with $\min (S) = r$ and $\max (S) = r+s$, there exists a connected degree-continuous graph $G$ with the degree set $S$ and diameter $2s+2$. The minimum order of a degree-continuous graph with a prescribed degree set is studied. Furthermore, it is shown that for every graph $G$ and convex set $S$ of positive integers containing the integer 2, there exists a connected degree-continuous graph $H$ with the degree set $S$ and containing $G$ as an induced subgraph if and only if $\max (S)\ge \Delta (G)$ and $G$ contains no $r-$regular component where $r = \max (S)$.
We consider, for a positive integer k, induced subgraphs in which each component has order at most k. Such a subgraph is said to be k-divided. We show that finding large induced subgraphs with this property is NP-complete. We also consider a related graph-coloring problem: how many colors are required in a vertex coloring in which each color class induces a k-divided subgraph. We show that the problem of determining whether some given number of colors suffice is NP-complete, even for 2-coloring a planar trianglefree graph. Lastly, we consider Ramsey-type problems where graphs or their complements with large enough order must contain a large k-divided subgraph. We study the asymptotic behavior of ''k-divided Ramsey numbers''. We conclude by mentioning a number of open problems.