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 ∈ L, the subgraph induced by its non-neighbors has a matching of cardinality |L| − 1. An `-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 `-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 `-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.