**Definition:**
A $\textit{matching}$ in a graph $G$ is a subgraph consisting of pairwise disjoint edges. If the subgraph is an induced subgraph, the matching is an $\textit{induced matching}$. The largest size of an induced matching in $G$ is called its induced matching number and denoted by $a(G)$.

Let $G$ be a bipartite graph. Is it possible to characterize the set of vertices $x$ of $G$ such that $a(G \setminus x)<a(G)$ ?

typeof characterization? @selva $\endgroup$ – Peter Heinig May 23 '17 at 7:42largestinduced matchings of $G$. Then $x\in V(G)$ is a vertex with the property $a(G\setminus x)<a(G)$ if and only if $x \in \bigcap_{\mathcal{M}\in\mathfrak{M}} (\cup\mathcal{M})$." This is a characterization of sorts. Since $\mathfrak{M}$ can be expected to typically have little humanly understandable structure (whatever that means), this is not an enlightening characterization, though. $\endgroup$ – Peter Heinig May 23 '17 at 8:03