Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing
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 $E$...
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| Veröffentlicht in: | TheoretiCS Jg. 1 |
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| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
TheoretiCS Foundation e.V
21.12.2022
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| Schlagworte: | |
| ISSN: | 2751-4838, 2751-4838 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | 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 $\pi:V\to V$ is proportional to the number of non-fixed-points of
$\pi$. 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 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. 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. |
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| ISSN: | 2751-4838 2751-4838 |
| DOI: | 10.46298/theoretics.22.1 |