Problem understanding "connectivity" characteristic for the $k$-connected subgraph problem
I think I figured it out.


$Com_{H}(v, u)$ are paths leaving the region and connecting through some vertices outside the region.

$Path_{H}(v)$ are the paths connecting $v$ to the vertices that are not included in this region.

$Path_{H}$ are paths inside the region



If this is true, then the proof of Lemma 3.2 has some typing errors (the 2 a.m. kind). And it should be:


  To compute $Com_{H}(v, u)$ where u, u v are non-portal vertices in $H$, for each competition $D$ in $Com_{H}(v, u)$, we insert D and the set difference $D\backslash B$ into $Com_{H \cup B}(v, u)$ (not $H \cup C$).
  [The rest is good]


I am not really sure, so please let me know what you think about this via comments \ up-votes \ telepathy.