TITLE: Proof of Gallai Theorem for factor critical graphs
QUESTION [6 upvotes]: I am trying to find the proof of the following theorem:

Definition 1.2. A vertex $v$ is essential if every maximum matching of $G$ covers $v$ (or $\nu(G-v) = \nu(G)-1$). It is avoidable if some maximum matching of $G$ exposes $v$ (or $\nu(G-v) = \nu(G)$). A graph $G$ is factor-critical if $G-v$ has a perfect matching for any $v \in V(G)$.
Theorem 1.3. Let $G$ be a connected graph. Then $G$ is factor-critical if and only if every vertex is avoidable.

I don't think this is something I can prove on my own. However, I can't seem to find the proof of it online either.
It makes sense intuitively. If for every vertex $v$, in graph $G$, there exists a perfect matching that doesn't include $v$, then removing $v$ and its incident edges would result in a perfect matching because one existed that didn't include $v$ in the first place.
Could someone provide a more formal proof for this theorem, or provide a link to where I might find the proof?
Thank You

REPLY [2 votes]: This can be proved quickly using the Tutte-Berge formula. For a subset of vertices $U \subseteq V$, let $q(U)$ denote the number of odd components of $G - U$. The Tutte-Berge formula states, in effect, that there is some $U \subseteq V$ such that a maximum size matching of $G$ avoids exactly $q(U) - |U|$ vertices. Now it is easy to see that any matching of $G$ avoids at least $q(U) - |U|$ vertices of $G-U$: from each of the $q(U)$ odd components at least one vertex must be either unmatched or matched to a vertex in $U$, and the latter can happen for at most $|U|$ vertices.
It follows that the vertices in $U$ are not avoidable, so by assumption $U$ must be empty. Then, since $G$ is connected, $q(U) - |U| = q(U)$ is either $0$ or $1$. The former would mean that $G$ has a perfect matching, so that none of the vertices are avoidable, a contradiction. So every maximum matching avoids exactly one vertex, which, together with the assumption that every vertex is avoidable, implies that $G$ is factor-critical.
In general, the behaviour of the set of avoidable vertices is described by the so-called Edmonds-Gallai decomposition of the graph, see e.g. Theorem 2 of these lecture notes. Gallai's Theorem immediately follows from this decomposition.