Block-Graph Cover Reduction Conjecture

Prove or disprove that whenever H is a block graph of a graph H', the H-Cover problem for simple input graphs polynomially reduces to the H'-Cover problem for simple input graphs.

Background

The paper develops reductions that transfer complexity results from selected block graphs of a target graph to the full target graph. In particular, the authors prove such reductions under additional structural conditions, including balancedness, and use them in the analysis of dangerous interblock graphs.

The concluding conjecture proposes that the same reduction principle should hold for every block graph H of every graph H', without the restrictions established in the paper. If true, it would substantially broaden the method for propagating NP-hardness and polynomial-time results among graph-cover instances.