I am working my way through "Matching Theory" by Lovasz and Plummer. Lemma 3.3.13 is where I am stuck.
Let $G$ be a graph such that $C(G)=\emptyset$ and let $X$ and $Y$ be barriers such that no line connects $X-Y$ to $Y-X$. Then $X\cap Y$ and $X\cup Y$ are barriers.
The proof starts with using Lemma 1.3.2, which states that in a bipartite graph $G=(A+B,E)$, for subsets $X,Y\subseteq A$ we have $def(X\cap Y)+def(X\cup Y) \geq def(X)+def(Y)$. (Deficiency in bipartite graph).
Apparently this lemma and the restrictions on $X,Y$ give us $c(G-X\cap Y)+c(G-X\cup Y) \geq c(G-X)+c(G-Y)$, where $c(G$ ist the number of connected components of G. But I don't see how.
Here is my thought process so far: We want to inbed the barriers $X$ and $Y$ in the bipartite graph we get using the Gallai-Edmonds Structure Theorem. Then we use the lemma and plug in the definition of deficiency in bipartite graphs ($def(A)=|A|-|\Gamma(A)|$, where $\Gamma(A)$ is the set of neighbours of A in G) to get $|\Gamma(X)|+|\Gamma(Y)|\geq |\Gamma(X\cap Y)|+|\Gamma(X\cup Y)|$. But how can we use this to get any implication on the components? My intuition would be that the inequality with the components should be the other way around because with the information above, we know that $X$ and $Y$ were connected to more components than $X\cap Y$ and $X\cup Y$, therefore we are dividing more components when we look at the residual Graph.