k-Majority Digraphs and the Hardness of Voting with a Constant Number of Voters

Bachmeier, Georg; Brandt, Felix; Geist, Christian; Harrenstein, Paul; Kardel, Keyvan; Peters, Dominik; Seedig, Hans Georg

doi:10.48550/arxiv.1704.06304

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Lemma 26 ([4]1

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2019

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“…By the way, we prove it by constructing a transitive digraph with the weak majority dimension greater than d for each positive integer d. Our result is also meaningful in a following sense. In 1941, Dushnik and Miller showed that a digraph has majority dimension at most two if and only if it is transitive and its incomparability graph is transitive orientable [1]. By looking at their result, one might think that a transitive digraph has a small majority dimension.…”

Section: Lemma 26 ([4]mentioning

confidence: 99%

On weak majority dimensions of digraphs

Eoh,

Kim

2019

Preprint

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In this paper, we introduce the notion of the weak majority dimension of a digraph which is well-defined for any digraph. We first study properties shared by the weak dimension of a digraph and show that a weak majority dimension of a digraph can be arbitrarily large. Then we present a complete characterization of digraphs of weak majority dimension 0 and 1, respectively, and show that every digraph with weak majority dimension at most two is transitive. Finally, we compute the weak majority dimensions of directed paths and directed cycles and pose open problems.

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Section: Lemma 26 ([4]mentioning

confidence: 99%