On the complexity of the identifiable subgraph problem, revisited

TL;DR

This paper investigates the computational complexity of identifying identifiable subgraphs in bipartite graphs, showing polynomial solvability for certain variants and W[1]-hardness for parameterized versions.

Contribution

It proves polynomial algorithms for the main problem and its deletion variant, and establishes W[1]-hardness for two parameterized variants, clarifying the problem's complexity landscape.

Findings

Polynomial-time algorithms for the identifiable subgraph problem.

W[1]-hardness results for specific parameterized variants.

Complemented existing APX-hardness results with new complexity insights.

Abstract

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.