On interval graphs, minimal vertex separators are well understood: they are cliques, there are no more than $n$ ones. However, when we turn to the minimal edge cut, my search found no even one single paper, which surprised me.
To make it more precise, the edge cuts are defined as follows. In an interval graph $G=(V,E)$, a pair of vertices $u$ and $v$ is called a dominating pair if there is a path $P$ between them such that all vertices of $V$ are adjacent to some vertex in $P$. A partition $(X,Y)$ of $V$ is called an edge cut if $uin X$ and $vin Y$.
A slight different way to define edge cuts is through the clique path decomposition. It is known that a graph is an interval graph if and only if there is a linear ordering of its $c$ maximal cliques in a way that each vertex appears in a consecutive subset of them. Then an edge cut is defined as a partition $(X,Y)$ where $K_1setminus K_2in X$ and $K_csetminus K_{c-1}in Y$.
These two definitions are not exactly the same, and may have different minimum cut sizes. But I believe they are related. Also note that it’s uninteresting to study the number of minimal edge cuts, as a clique is an interval graph where each partition decides a different minimal edge cut.
EDIT. My question: what’s the properties of the minimum edge cuts?