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1. Irreducibility of Recombination Markov Chains in the Triangular Lattice

   Cannon, Sarah ( May 2023 , SIAM Conference on Applied and Computational Discrete Algorithms (ACDA23))

   Berry, Jonathan ; Shmoys, David ; Cowen, Lenore ; Naumann, Uwe (Ed.)

   In the United States, regions (such as states or counties) are frequently divided into districts for the purpose of electing representatives. How the districts are drawn can have a profound effect on who's elected, and drawing the districts to give an advantage to a certain group is known as gerrymandering. It can be surprisingly difficult to detect when gerrymandering is occurring, but one algorithmic method is to compare a current districting plan to a large number of randomly sampled plans to see whether it is an outlier. Recombination Markov chains are often used to do this random sampling: randomly choose two districts, consider their union, and split this union up in a new way. This approach works well in practice and has been widely used, including in litigation, but the theory behind it remains underdeveloped. For example, it's not known if recombination Markov chains are irreducible, that is, if recombination moves suffice to move from any districting plan to any other. Irreducibility of recombination Markov chains can be formulated as a graph problem: for a planar graph G, is the space of all partitions of G into κ connected subgraphs (κ districts) connected by recombination moves? While the answer is yes when districts can be as small as one vertex, this is not realistic in real-world settings where districts must have approximately balanced populations. Here we fix district sizes to be κ1 ± 1 vertices, κ2 ± 1 vertices,… for fixed κ1, κ2,…, a more realistic setting. We prove for arbitrarily large triangular regions in the triangular lattice, when there are three simply connected districts, recombination Markov chains are irreducible. This is the first proof of irreducibility under tight district size constraints for recombination Markov chains beyond small or trivial examples. The triangular lattice is the most natural setting in which to first consider such a question, as graphs representing states/regions are frequently triangulated. The proof uses a sweep-line argument, and there is hope it will generalize to more districts, triangulations satisfying mild additional conditions, and other redistricting Markov chains. 

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2. Reconfiguration of Connected Graph Partitions via Recombination

   https://doi.org/10.1007/978-3-030-75242-2_4  

   Akitaya, Hugo A. ; Korman, Matias ; Korten, Oliver ; Souvaine, Diane L. ; Toth, Csaba D. ( May 2021 , Algorithms and Complexity (CIAC 2021))

   null (Ed.)

   Motivated by applications in gerrymandering detection, we study a reconfiguration problem on connected partitions of a connected graph G. A partition of V(G) is connected if every part induces a connected subgraph. In many applications, it is desirable to obtain parts of roughly the same size, possibly with some slack s. A Balanced Connected k-Partition with slack s, denoted (k, s)-BCP, is a partition of V(G) into k nonempty subsets, of sizes n1,…,nk with |ni−n/k|≤s , each of which induces a connected subgraph (when s=0 , the k parts are perfectly balanced, and we call it k-BCP for short). A recombination is an operation that takes a (k, s)-BCP of a graph G and produces another by merging two adjacent subgraphs and repartitioning them. Given two k-BCPs, A and B, of G and a slack s≥0 , we wish to determine whether there exists a sequence of recombinations that transform A into B via (k, s)-BCPs. We obtain four results related to this problem: (1) When s is unbounded, the transformation is always possible using at most 6(k−1) recombinations. (2) If G is Hamiltonian, the transformation is possible using O(kn) recombinations for any s≥n/k , and (3) we provide negative instances for s≤n/(3k) . (4) We show that the problem is PSPACE-complete when k∈O(nε) and s∈O(n1−ε) , for any constant 0<ε≤1 , even for restricted settings such as when G is an edge-maximal planar graph or when k≥3 and G is planar. 

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3. Combinatorics of exceptional sequences in type A

   Garver, Alexander ; Igusa, Kiyoshi ; Matherne, Jacob P. ; Ostroff, Jonah ( February 2019 , The Electronic journal of combinatorics)

   Exceptional sequences are certain sequences of quiver representations. We introduce a class of objects called strand diagrams and use these to classify exceptional sequences of representations of a quiver whose underlying graph is a type An Dynkin diagram. We also use variations of these objects to classify $c$-matrices of such quivers, to interpret exceptional sequences as linear extensions of explicitly constructed posets, and to give a simple bijection between exceptional sequences and certain saturated chains in the lattice of noncrossing partitions. 

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4. Subdivided Claws and the Clique-Stable Set Separation Property

   https://doi.org/10.1007/978-3-030-62497-2  

   Chudnovsky, Maria and ( February 2021 , MATRIX book series)

   null (Ed.)

   Let C be a class of graphs closed under taking induced subgraphs. We say that C has the clique-stable set separation property if there exists c ∈ N such that for every graph G ∈ C there is a collection P of partitions (X, Y ) of the vertex set of G with |P| ≤ |V (G)| c and with the following property: if K is a clique of G, and S is a stable set of G, and K ∩ S = ∅, then there is (X, Y ) ∈ P with K ⊆ X and S ⊆ Y . In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by M. G¨o¨os in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant c exists. In this paper we define two infinite families S, K of graphs and show that for every S ∈ S and K ∈ K, the class of graphs with no induced subgraph isomorphic to S or K has the clique-stable set separation property. 

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5. Subdivided Claws and the Clique-Stable Set Separation Property

   https://doi.org/10.1007/978-3-030-62497-2_29  

   Chudnovsky, Maria and ( February 2021 , MATRIX book series)

   Let C be a class of graphs closed under taking induced subgraphs. We say that C has the clique-stable set separation property if there exists c∈N such that for every graph G∈C there is a collection P of partitions (X,Y) of the vertex set of G with |P|≤|V(G)|c and with the following property: if K is a clique of G, and S is a stable set of G, and K∩S=∅, then there is (X,Y)∈P with K⊆X and S⊆Y. In 1991 M. Yannakakis conjectured that the class of all graphs has the clique-stable set separation property, but this conjecture was disproved by Göös in 2014. Therefore it is now of interest to understand for which classes of graphs such a constant c exists. In this paper we define two infinite families S,K of graphs and show that for every S∈S and K∈K, the class of graphs with no induced subgraph isomorphic to S or K has the clique-stable set separation property. 

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