We characterize chordal graphs.1
Let $(V, E)$ be an undirected graph. A set $S \subset V$ is a vertex separator for two vertices $v, w$ (or a $vw$-separator) if $v$ and $w$ are disconnected in the subgraph induced by $V \setminus S$. There always exists a vertex separator for two nonadjacent vertices.
A vertex separator is a minimal vertex separator for two vertices if no proper subset of it is a vertex separator for those vertices. Another term for vertex separator is cutset. Similarly, one for minimal vertex separator is relatively minimal cutset.
For example, for the graph in Figure below , $\set{c,e}$ is a minimal $ag$-separator and $\set{b,c,e}$ is a minimal $ad$-separator. A minimal vertex separator may contain another minimal vertex separator if they are minimal for different pairs of vertices.