On the complexity of the identifiable subgraph problem, revisited

A bipartite graph $G=(L,R;E)$ with at least one edge is said to be identifiable if for every vertex $v\in L$, the subgraph induced by its non-neighbors has a matching of cardinality $|L|-1$. An $\ell$-subgraph of $G$ is an induced subgraph of $G$ obtained by deleting from it some vertices in $L$ together with all their neighbors. The Identifiable Subgraph problem is the problem of determining whether a given bipartite graph contains an identifiable $\ell$-subgraph. We show that the Identifiable Subgraph problem is polynomially solvable, along with the version of the problem in which the task is to delete as few vertices from $L$ as possible together with all their neighbors so that the resulting $\ell$-subgraph is identifiable. We also complement a known APX-hardness result for the complementary problem in which the task is to minimize the number of remaining vertices in $L$, by showing that two parameterized variants of the problem are W[1]-hard.