Bicolored graph partitioning, or: gerrymandering at its worst

Abstract

This study is motivated by an electoral application where we look into the following question: how much biased can the assignment of parliament seats be in a majority system under the effect of vicious gerrymandering when the two competing parties have the same electoral strength? To give a first theoretical answer to this question, we introduce a stylized combinatorial model, where the territory is represented by a rectangular grid graph, the vote outcome by a "balanced" red/blue node bicoloring and a district map by a connected partition of the grid whose components all have the same size. We constructively prove the existence in cycles and grid graphs of a balanced bicoloring and of two antagonist "partisan" district maps such that the discrepancy between their number of "red" (or "blue") districts for that bicoloring is extremely large, in fact as large as allowed by color balance. © 2009 Elsevier B.V. All rights reserved

A discrete districting plan

“…the case where the 14 and 4 weight vertexes are grouped together. This 4-partition has energy F λ (C 4 ) = 36λ + 2 √ 17(1 − λ) , 1 The choice λ = 1 2 has nothing special. For any λ ∈ (0, 1) the weights of the graph Γ can be suitably chosen to have the same behavior of the shown example: it is sufficient to multiply the weights on vertexes by 1 2λ and those on edges by Figure 5.…”

The efficiency of areal units in spatial analysis: Assessing the performance of functional and administrative regions

“…This is not always the case, however, and in some cases administrative systems are not well designed for political reasons or just because the rules or norms are unsuitable or they are designed on purpose (as is the case of Slovakia: see for example Buček, 2002Buček, , 2005or Romania: Suciu, 2002 Bunge, 1966;Johnston, 2002;Moore, 2002;Suciu, 2002;Apollonio et al, 2009 In theory, the definition of administrative regions should respect three basic principles with regard to a space: spatial efficiency, spatial equity, and spatial stability (Bezák, 1997, who builds upon the concepts put forward by Goodall, 1987;Michniak, 2003;Halás and Klapka, 2012;Klapka, et al, 2014). The principle of spatial efficiency states that the administrative geography of a territory should reflect the population distribution and its spatial behaviours (particularly spatial movements) to the greatest possible extent.…”

The Complexity of Gerrymandering Over Graphs: Paths and Trees

“…We remark that all our results except for Theorem 1 (that is, the NP-hardness on paths) also transfer to the slightly different model of Cohen-Zemach et al [4]. 1 We also use an equivalent interpretation of solution partitions V for Gerrymandering over Graphs. Since each part V i ∈ V has to induce a connected subgraph, in the spirit of edge deletion problems from algorithmic graph theory, we also represent solutions by a set of edges such that removing these yields the disjoint union of subgraphs induced by each part V i ∈ V. In Figure 1, removing the edges {w, x} and {w, y} yields a solution.…”

“…We mention in passing that earlier work also studied the special case of gerrymandering on grid graphs. More specifically, Apollonio et al [1] analyzed gerrymandering in grid graphs where each district in the solution has to be of (roughly) the same size and they analyzed, focusing on two candidates (equivalently, two parties), the maximum possible win margin if the two candidates had the same amount of support. Later, Borodin et al [2] also considered gerrymandering on grid graphs with two parties (expressed by colors red and blue), but here each vertex represents a polling station and thus is partially "red" and partially "blue" colored.…”