cut

let $\text{[math]}$ be a connected undirected graph. assume that $\text{[math]}$ are subsets of $\text{[math]}$ such that $\text{[math]}$ , the partitioning into groups $\text{[math]}$ by removing edges or vertices that connect them is called a cut in $\text{[math]}$ , and if applied $\text{[math]}$ becomes a disconnected graph.

note that although we state that a cut partitions a graph into two groups $\text{[math]}$ , the subgraphs restricted to $\text{[math]}$ dont necessarily have to be connected, so a cut may result in more than two disconnected subgraphs, for otherwise we call it a broken link: blk:simple cut (see broken link: blk:fig-cut-1).

for a graph $\text{[math]}$ and a set $\text{[math]}$ of vertices of $\text{[math]}$ , we define $\text{[math]}$ to be the set of edges having one endpoint in $\text{[math]}$ and one endpoint not in $\text{[math]}$ . we say a set of edges of $\text{[math]}$ is a cut of $\text{[math]}$ if it has the form $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 chapter 3.2.1 undirected cuts]

we define $\text{[math]}$ to be the set of arcs whose tails are in $\text{[math]}$ and whose heads are not in $\text{[math]}$ . note that $\text{[math]}$ is a subset of $\text{[math]}$ . a set of arcs of $\text{[math]}$ is a directed cut (a.k.a dicut) if it has the form $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 chapter 3.2.2 directed dicuts]

the cut set of a cut $\text{[math]}$ in a graph $\text{[math]}$ is the set of edges whose endpoints each exist in a different subset of the vertices that would result from the cut:
$\text{[math]}$ where $\text{[math]}$ are the vertex subsets after the cut

for a set $\text{[math]}$ of vertices of $\text{[math]}$ , we define $\text{[math]}$ to be the set of darts whose tails are in $\text{[math]}$ and whose heads are not in $\text{[math]}$ . note that $\text{[math]}$ has one dart for each edge in $\text{[math]}$ . we refer to $\text{[math]}$ as the dart boundary of $\text{[math]}$ in $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 chapter 3.2.3 dart cuts]

let $\text{[math]}$ be a graph, let $\text{[math]}$ be a connected component of $\text{[math]}$ , and let $\text{[math]}$ be a subset of the vertices of $\text{[math]}$ . we say a cut $\text{[math]}$ or a dart cut $\text{[math]}$ is a simple cut if $\text{[math]}$ is connected in $\text{[math]}$ and $\text{[math]}$ is connected in $\text{[math]}$ . (some authors have used the term bond for this concept.)
note that a cut in $\text{[math]}$ is minimal among nonempty cuts iff it is a simple cut. any cut can be written as a disjoint union of simple cuts.
[cite:;taken from @graph_klein_2024 chapter 3.2.4 simple cuts]

broken link: xopp-figure:/home/mahmooz/brain/pen/1732965636.577574.xopp

let $\text{[math]}$ be a graph, and let $\text{[math]}$ be a spanning forest of $\text{[math]}$ . for a tree edge $\text{[math]}$ , let $\text{[math]}$ be the connected component of $\text{[math]}$ that contains $\text{[math]}$ . for $\text{[math]}$ , let
$\text{[math]}$ we call $\text{[math]}$ the fundamental cut of $\text{[math]}$ in $\text{[math]}$ with respect to $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 chapter 3.2.5 tree edges and fundamental cuts]

let $\text{[math]}$ be a connected planar embedded graph with spanning tree $\text{[math]}$ and let $\text{[math]}$ be an edge of $\text{[math]}$ . then the darts of the fundamental cut of $\text{[math]}$ in $\text{[math]}$ with respect to $\text{[math]}$ are the same as the darts of the fundamental cycle of $\text{[math]}$ in $\text{[math]}$ with respect to $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 lemma 4.6.1]

the drawing in broken link: blk:fig-fund-cut-cycle-duality demonstrates this idea concretely which may help intuition.
broken link: xopp-figure:/home/mahmooz/brain/pen/1733339937.1868992.xopp
broken link: xopp-figure:/home/mahmooz/brain/pen/1734679621.9722865.xopp