# American Institute of Mathematical Sciences

December  2019, 14(4): 771-788. doi: 10.3934/nhm.2019031

## A discrete districting plan

 1 Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 53/A, I-43124 Parma, Italy 2 Dipartimento di Matematica, Università di Pavia, via Ferrata 5, I-27100 Pavia, Italy

* Corresponding author: G. Saracco

Received  January 2019 Revised  June 2019 Published  October 2019

Fund Project: A. Saracco was partially supported by INdAM-GNSAGA. G. Saracco was partially supported by the INdAM-GNAMPA 2019 project "Problemi isoperimetrici in spazi Euclidei e non"

The outcome of elections is strongly dependent on the districting choices, making thus possible (and frequent) the gerrymandering phenomenon, i.e. politicians suitably changing the shape of electoral districts in order to win the forthcoming elections. While so far the problem has been treated using continuous analysis tools, it has been recently pointed out that a more reality-adherent model would use the discrete geometry of graphs or networks. Here we propose a parameter-dependent discrete model for choosing an "optimal" districting plan. We analyze several properties of the model and lay foundations for further analysis on the subject.

Citation: Alberto Saracco, Giorgio Saracco. A discrete districting plan. Networks & Heterogeneous Media, 2019, 14 (4) : 771-788. doi: 10.3934/nhm.2019031
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Removing or adding a vertex; numbers correspond to the weight of the vertexes; all edges are supposed to have weight $1$
Removing or adding an edge; numbers correspond to the weight of the related vertexes and edges
Splitting two vertexes is not always possible by simply modifying the weight of their common edge
A graph where the optimal $4$-partition is not a $2$-refining of the optimal $2$-partition
A graph where the optimal $4$-partition is a $2$-refining of the optimal $2$-partition, but does not induce the optimal $2$-partitions on its components
A graph where the optimal $4$-partition is not a $2$-refining of the optimal $2$-partition for suitable choices of $\alpha = \alpha(p)$
A graph where the optimal $4$-partition is not a $2$-refining of the optimal $2$-partition
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