Strongly consistent ordering with a given partition → Proper thinness with an input class budget Complexity open

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Why this question matters

The question isolates the complexity contributed by choosing the partition in an ordering-based graph representation.

Strong consistency with a supplied partition differs from finding both a partition and an order under a class budget.

The reduction to construct

The source gives a graph and a vertex partition. Its outputs are orders for which both the order and its reverse are consistent with the partition: if u<v<w, u and v share a class, and uw is an edge, then vw is an edge. Otherwise return NO-SOLUTION.

The source supplies a graph and vertex partition and asks for an order strongly consistent in both directions. The target supplies only a graph and a class budget, and asks for both a partition within that budget and such an order.

Construct deterministic polynomial-time maps F and G: F sends every legal source instance to a legal target instance, and G(x,y) is a valid source output for every valid output y of F(x). Preserve the stated threshold, domain and promises. The requested complexity conclusion is NP-hardness for the stated target problem.

The target may choose a different partition; forcing the intended classes without fixing them externally is the missing step.

What a complete answer must include

Give explicit construction and recovery algorithms, a general proof for every legal input and every valid target output, and polynomial runtime and encoding-size bounds. Specify finite output encodings and handle NO-SOLUTION outputs when applicable. Tests compare recovered source outputs with independent source solutions.

References

Literature checked 2026-09-16. This summarizes the archived literature search on the date above. Unpublished, unindexed and overlooked work remains outside coverage; no new novelty assessment was performed.