# Konig's Theorem for Min Weight Vertex Cover?

Koning's theorem states that the cardinality of the maximum matching in a bipartite graph is equal to the size of its minimum vertex cover.

Wikipedia states that there is an equivalent version of the theorem for weighted graphs as well. I tried searching in a lot of places but could not find this theorem.

To be more specific, I want to know what is the equivalent version of Konig's theorem in the case of Minimum Weight Vertex Cover in Bipartite Graphs where each vertex $v \in G$ is assigned a weight $w_v$ and the task is to find a vertex cover of the minimum possible weight.

If the equivalence is just the maximum weighted matching in $G$, what are the weights of the edges in this graph? Because as far as I understand, we have only assigned weights to the vertices of the graph $G$.

Thank you for your time :)

Let $G = (A,B)$ be the given bipartite graph. Construct a flow network $N$ by connecting the a source $S$ to all nodes in $A$ and all nodes in $B$ to a sink $T$.
Let the capacity of all original edges in $G$ be $\infty$. The capacity of all edges of the form $(S,a)$ where $a \in A$ will be $w_a$ and the similarly the capacity of all edges $(b, T)$ where $b \in B$ will be $w_b$. Every $S-T$ cut in this network corresponds to exactly one vertex cover and every vertex cover corresponds to an $S-T$ cut. Thus the min-cut a.k.a. the maximum flow will give us the minimum weight vertex cover.