An edge $e$ of a $k$-connected graph $G$ is said to be $k$-contractible (or simply contractible) if the graph obtained from $G$ by contracting $e$ (i.e., deleting $e$ and identifying its ends, finally, replacing each of the resulting pairs of double edges by a single edge) is still $k$-connected. In 2002, Kawarabayashi proved that for any odd integer $k\geq 5$, if $G$ is a $k$-connected graph and $G$ contains no subgraph $D=K_{1}+(K_{2}\cup K_{1, 2})$, then $G$ has a $k$-contractible edge. In this paper, by generalizing this result, we prove that for any integer $t\geq 3$ and any odd integer $k \geq 2t+1$, if a $k$-connected graph $G$ contains neither $K_{1}+(K_{2}\cup K_{1, t})$, nor $K_{1}+(2K_{2}\cup K_{1, 2})$, then $G$ has a $k$-contractible edge.
Let $p$ be a prime. We assign to each positive number $k$ a digraph $G_{p}^{k}$ whose set of vertices is $\{1,2,\ldots ,p-1\}$ and there exists a directed edge from a vertex $a$ to a vertex $b$ if $a^k\equiv b \pmod {p}$. In this paper we obtain a necessary and sufficient condition for $G_{p}^{k_{1}}\simeq G_{p}^{k_{2}}$.
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.