This is a trivial question, but I can't seem to get a clear answer. Could someone please explain to me what is meant by the phrase "cut induced by". For example, in the following (see note 4)
I understand what a dual graph is, and I understand what $f^*$ and $G^*$ are supposed to represent. I'm having trouble understanding $E_f^*$ because I don't understand what "cut induced by" means.
Given a graph $G$ with vertex set $V$, let $U$ and $W$ form a partition of $V$. Then the cut induced by this partition is formally defined as the set of all the edges in $G$ that joins a vertex in $U$ with a vertex in $W$. So if $v$ is any vertex, then the cut induced by $v$ in $G$ refers to the cut induced by the partition $\{v\}$ and $V\setminus \{v\}$ of $V$ i.e., it is the set of all the edges adjacent to $v$.
Since $f^*$ is a vertex in $G^*$, so cut induced by $f^*$ would be the set of all the edges adjacent to $f^*$ in $G^*$.