Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing
A graph G is called {\em 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=(VE) is {\em 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 :VV is proportional to the number of non-fixed-points of . 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, on {\em pairs} of vertices of the original graph, is expanding. In this case the original graph is (not only robustly self-ordered but) also expanding. The second construction proceeds in three steps: It boosts more »
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Publication Date:
NSF-PAR ID:
10273422
Journal Name:
Electronic colloquium on computational complexity
Volume:
ECCC TR20-149
ISSN:
1433-8092