Year: 2024
Type: article
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 irreducible 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.
Source: Annals of Combinatorics
Author: Vincent Pilaud
Institution: Universitat de Barcelona
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,790
Awards: G3219416326, G8757075480
OpenAlex ID: W4226055348