Strong Dichotomy Conjecture for Graph Covers

Establish that for every graph H, the H-Cover problem is either polynomial-time solvable for arbitrary input graphs or NP-complete when restricted to simple input graphs.

Background

The paper studies the computational complexity of the fixed-target graph covering problem H-Cover for highly general colored mixed multigraphs, including loops, multiple edges, directed edges, and semi-edges. A principal objective is to determine whether every target graph exhibits a strong polynomial-time/NP-complete dichotomy.

The conjecture is motivated by the observation that all previously known NP-hard instances of H-Cover remain NP-hard even when the input graphs are simple. The paper proves this property for all target graphs whose degree-partition blocks have size at most two, but leaves the assertion open for arbitrary target graphs.