If I understand correctly, DBSCAN is the following method of decomposing a set, which has parameters $\epsilon$ and $k$. Given a set $E$ of points in $\mathbb R^n$, consider the graph $G$ whose vertices are the points of $E$ and where there is an edge between $x$ and $y$ whenever $|x-y| \leq \epsilon$. Consider the subgraph $G'$ consisting of vertices of $E$ of degree at least $k$, and let $C'_1, ... ,C'_N$ be the connected components of $G'$. Let $C_1, ... , C_N$ be subsets of $G$ where a point is in $C_i$ if either it is in $C'_i$ or it is a neighbor of a point in $C'_i$. A point can be in more than one $C_i$. Since a point can be in $C_i$ for more than one value of $i$, if you want the $C_i$ to be mutually disjoint, you have to enforce this by, for example, arbitrarily assigning each point to exactly one of the $C_i$ it belongs to. Points which are not in any $C_i$ are called "noise".
This algorithm seems to have a third parameter $s$ which is not usually stated and is by default set to $1$. This is the number of edges which must be traversed to get from a boundary point to a core point, where $C'_i$ is the set of core points and $C_i \setminus C'_i$ is the set of boundary points. What happens if $s=0$? Or $s=2$? That is, what if we just set $C_i = C'_i$, or what if we say $p \in C'_i$ if there exist $q, r$ such that $p$ neighbors $q$, $q$ neighbors $r$, and $r \in C_i$? Are variations where $s$ is manipulated studied?
One issue with this variation is that if $s=2$, a core point of one cluster could also be a boundary point of another cluster.
