Acyclic reorientation lattices and their lattice quotients

Abstract

We prove that the acyclic reorientation poset of a directed acyclic graph D is a lattice if and only if the transitive reduction of any induced subgraph of D is a forest. We then show that the acyclic reorientation lattice is always congruence normal, semidistributive (thus congruence uniform) if and only if D is filled, and distributive if and only if D is a forest. When the acyclic reorientation lattice is semidistributive, we introduce the ropes of D that encode the join irreducibles acyclic reorientations and exploit this combinatorial model in three directions. First, we describe the canonical join and meet representations of acyclic reorientations in terms of non-crossing rope diagrams. Second, we describe the congruences of the acyclic reorientation lattice in terms of lower ideals of a natural subrope order. Third, we use Minkowski sums of shard polytopes of ropes to construct a quotientope for any congruence of the acyclic reorientation lattice.

Combinatorial Gray Codes — an Updated Survey

“…• Flip graphs on acyclic orientations of a graph induced by flipping the orientation of a single arc also have a poset structure, and they can be realized as polytopes [SZ98]. These posets sometimes form a lattice and admit lattice congruences that can also be realized as polytopes [Pil21], generalizing the aforementioned quotientopes.…”

“…Pilaud [Pil21] equipped G(H) with a poset structure, and found conditions on H for when this poset is a lattice with various additional properties. P56 He also describes the corresponding lattice congruences and realizations of their quotients as polytopes, and he raises the question whether they admit a Hamilton cycle.…”

Combinatorial Gray codes-an updated survey

“…Pilaud [Pil21] equipped G(H) with a poset structure, and found conditions on H for when this poset is a lattice with various additional properties.…”