connected-components tree

let $\text{[math]}$ be a connected triangulated undirected graph. let $\text{[math]}$ be a vertex of $\text{[math]}$ , and let $\text{[math]}$ be a face incident to $\text{[math]}$ . for a face $\text{[math]}$ of $\text{[math]}$ , define its level to be the minimum level of a vertex incident to $\text{[math]}$ . thus, $\text{[math]}$ , and all faces to which $\text{[math]}$ is incident is also 0.
for an integer $\text{[math]}$ , let $\text{[math]}$ denote the set of faces of $\text{[math]}$ with level at least $\text{[math]}$ . let $\text{[math]}$ denote the set of connected components of the subgraph of $\text{[math]}$ induced by $\text{[math]}$ . we refer to a connected component $\text{[math]}$ as a level- $\text{[math]}$ component, and we define $\text{[math]}$ . with a slight abuse of notation we identify a connected component $\text{[math]}$ of $\text{[math]}$ , which is a subgraph of $\text{[math]}$ , with the set of vertices of $\text{[math]}$ (faces of $\text{[math]}$ ) that belong to $\text{[math]}$ . thus, when it is convenient, we regard $\text{[math]}$ as a subset of $\text{[math]}$ .
[cite:;taken from @graph_klein_2024 chapter 5.7 the connected-components tree]

for any level $\text{[math]}$ and any component $\text{[math]}$ , $\text{[math]}$ is a simple cut.
[cite:;taken from @graph_klein_2024 chapter 5.7 lemma 5.7.7]