Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing

Authors Oded Goldreich , Avi Wigderson

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Oded Goldreich
  • Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel
Avi Wigderson
  • School of Mathematics, Institute for Advanced Study, Princeton, USA


We are grateful to Eshan Chattopadhyay for discussions regarding non-malleable two-source extractors, and to Dana Ron for discussions regarding tolerant testing. We are deeply indebted to Jyun-Jie Liao for pointing out an error in the proof of [O. Goldreich and A. Wigderson, 2020].

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Oded Goldreich and Avi Wigderson. Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 12:1-12:74, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


A graph G is called self-ordered (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 self-ordered if the size of the symmetric difference between E and the edge-set of the graph obtained by permuting V using any permutation π:V → V is proportional to the number of non-fixed-points of π. In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs. We show that robustly self-ordered bounded-degree 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 polynomial-time (i.e., in time that is poly-logarithmic 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 self-ordering, which requires that an auxiliary graph is expanding. The second construction is iterative, boosting the property of robust self-ordering from smaller to larger graphs. Structuraly, the first construction always yields expanding graphs, while the second construction may produce graphs that have many tiny (sub-logarithmic) 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 non-malleable two-source extractors (with very weak parameters but with some additional natural features). We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree 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 bounded-degree graph model.

Subject Classification

ACM Subject Classification
  • Theory of computation → Generating random combinatorial structures
  • Asymmetric graphs
  • expanders
  • testing graph properties
  • two-source extractors
  • non-malleable extractors
  • coding theory
  • tolerant testing
  • random graphs


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