Abstract
A graph G is called selfordered (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from G to any graph that is isomorphic to G. We say that G = (V,E) is robustly selfordered if the size of the symmetric difference between E and the edgeset of the graph obtained by permuting V using any permutation π:V → V is proportional to the number of nonfixedpoints of π. In this work, we initiate the study of the structure, construction and utility of robustly selfordered graphs.
We show that robustly selfordered boundeddegree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomialtime (i.e., in time that is polylogarithmic in the size of the graph).
We provide two very different constructions, in tools and structure. The first, a direct construction, is based on proving a sufficient condition for robust selfordering, which requires that an auxiliary graph is expanding. The second construction is iterative, boosting the property of robust selfordering from smaller to larger graphs. Structuraly, the first construction always yields expanding graphs, while the second construction may produce graphs that have many tiny (sublogarithmic) connected components.
We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree) exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of nonmalleable twosource extractors (with very weak parameters but with some additional natural features).
We demonstrate that robustly selfordered boundeddegree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the boundeddegree and the dense graph models. Indeed, their robustness offers efficient, local and distance preserving reductions from testing problems on ordered structures (like sequences) to the unordered (effectively unlabeled) graphs. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the boundeddegree graph model.
BibTeX  Entry
@InProceedings{goldreich_et_al:LIPIcs.CCC.2021.12,
author = {Goldreich, Oded and Wigderson, Avi},
title = {{Robustly SelfOrdered Graphs: Constructions and Applications to Property Testing}},
booktitle = {36th Computational Complexity Conference (CCC 2021)},
pages = {12:112:74},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771931},
ISSN = {18688969},
year = {2021},
volume = {200},
editor = {Kabanets, Valentine},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14286},
URN = {urn:nbn:de:0030drops142867},
doi = {10.4230/LIPIcs.CCC.2021.12},
annote = {Keywords: Asymmetric graphs, expanders, testing graph properties, twosource extractors, nonmalleable extractors, coding theory, tolerant testing, random graphs}
}
Keywords: 

Asymmetric graphs, expanders, testing graph properties, twosource extractors, nonmalleable extractors, coding theory, tolerant testing, random graphs 
Collection: 

36th Computational Complexity Conference (CCC 2021) 
Issue Date: 

2021 
Date of publication: 

08.07.2021 