graph theory – Is there any theorem similar to the Tutte–Berge formula?

Tutte–Berge formula is a characterization of the size of a maximum matching in a graph.

The theorem states that the size of a maximum matching of a graph
${displaystyle G=(V,E)}$ equals $${displaystyle {frac {1}{2}}min _{Usubseteq V}left(|V|-(operatorname {odd} (G-U)-|U|)right),}$$ where
${displaystyle operatorname {odd} (H)}$ counts how many of the connected components of the graph ${displaystyle H}$ have an odd number of vertices.

And we can see from the formula that it’s independent of even connected components of $G-U$.

I wonder if there are any other graph parameters related to even connected components of $G-U$ or connected components of $G-U$. Indeed, it doesn’t have to be in the form of $G-U$.