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Separability Properties of Monadically Dependent Graph Classes
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Separability Properties of Monadically Dependent Graph Classes
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Paper
Separability Properties of Monadically Dependent Graph Classes
2025
Overview
A graph class \\( C\\) is monadically dependent if one cannot interpret all graphs in colored graphs from \\( C\\) using a fixed first-order interpretation. We prove that monadically dependent classes can be exactly characterized by the following property, which we call flip-separability: for every \\(rın N\\), \\(>0\\), and every graph \\(Gın C\\) equipped with a weight function on vertices, one can apply a bounded (in terms of \\(C,r,\\)) number of flips (complementations of the adjacency relation on a subset of vertices) to \\(G\\) so that in the resulting graph, every radius-\\(r\\) ball contains at most an \\(\\)-fraction of the total weight. On the way to this result, we introduce a robust toolbox for working with various notions of local separations in monadically dependent classes.
Publisher
Cornell University Library, arXiv.org
Subject
