# 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 $$G-U$$.