I’m sorry to bother you with this simple question. I would like to model a simple model of the minimum cover vertex set problem. I believe that the original problem is such as
$$ \min \quad \sum_{v\in V} x_v $$ subject to $$ x_u + x_v \ge 1 \quad \forall (u,v) \in E $$ $x_v$ is a binary variable, which takes value $1$ if vertex $v$ is included in the minimum set and $0$ otherwise. I have added another binary variable $z_{uv}$ with the following constraint $$ \sum_{(u,v)\in E} z_{uv} \ge k $$ where $k$ is a parameter and $z_{uv}$ equals $1$ if vertex $v$ is in the set while vertex $u$ is not a member of the minimum set.
So, $x$ is variable controling the vertices while $z$ controls the edges.
I hope you can help me with modeling the second constraint.
I’m confused about the interaction between if nodes are decided to be in the set (i.e., $x_v = 1$) and the edge connected to it. P.S the edges are defined as $e=(u,v)$ where $u,v$ are the vertices incident to that edge.
Thank you so much in advance.