# Testing Hereditary Properties of Ordered Graphs and Matrices

@article{Alon2017TestingHP,
title={Testing Hereditary Properties of Ordered Graphs and Matrices},
author={Noga Alon and Omri Ben-Eliezer and Eldar Fischer},
journal={2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)},
year={2017},
pages={848-858}
}
• Published 7 April 2017
• Mathematics
• 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
We consider properties of edge-colored vertex-ordered graphs} &#x2013; graphs with a totally ordered vertex set and a finite set of possible edge colors &#x2013; showing that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of two-dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first…
• Mathematics, Computer Science
• 2018
The theory of graph limits is extended to the ordered setting, presenting a limit object for dense vertex-ordered graphs, which is called an orderon, and it is shown that the space of orderons is compact with respect to this distance notion,Which is key to a successful analysis of combinatorial objects through their limits.
• Mathematics
Combinatorial Theory
• 2022
. A recent result of Alon, Ben-Eliezer and Fischer establishes an induced removal lemma for ordered graphs. That is, if F is an ordered graph and ε > 0 , then there exists δ F ( ε ) > 0 such that
The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant
• Mathematics
ArXiv
• 2018
The theory of graph limits is extended to the ordered setting, presenting a limit object for dense vertex-ordered graphs, which is called an orderon, and it is shown that the space of orderons is compact with respect to this distance notion,Which is key to a successful analysis of combinatorial objects through their limits.
• Mathematics, Computer Science
ITCS
• 2021
An ordered analogue of the well-known result by Alon and Stav on the furthest graph from a hereditary property is proved, the first known result of this type in the ordered setting, and an alternative analytic proof of the ordered graph removal lemma is described.
• Mathematics, Computer Science
Electron. Colloquium Comput. Complex.
• 2018
A wide class of properties of ordered structures - the earthmover resilient (ER) properties - are identified and it is shown that the "good behavior" of such properties allows us to obtain general testability results that are similar to (and more general than) those of unordered graphs.
• Mathematics
• 2019
The emerging theory of graph limits exhibits an analytic perspective on graphs, showing that many important concepts and tools in graph theory and its applications can be described more naturally
• Mathematics
APPROX-RANDOM
• 2017
It is shown that there exist hereditary properties of sequences that cannot be tested with sublinear queries, resolving an open question posed by Newman et al.
• Mathematics, Computer Science
APPROX-RANDOM
• 2017
This work establishes much more efficient removal lemmas for several special cases of the above problem and combines their efficient conditional regularity lemma for matrices with additional combinatorial and probabilistic ideas.

## References

SHOWING 1-10 OF 54 REFERENCES

• Mathematics
40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
• 1999
This theorem is used to prove that first order graph properties not containing a quantifier alternation of type "/spl forall//spl exist/" are always testable, while it is shown that some properties containing this alternation are not.
• Mathematics, Computer Science
STOC '05
• 2005
It is shown that any monotone graph property can be tested with one-sided error, and with query complexity depending only on ε, and this result implies the testability of well-studied graph properties that were previously not known to be testable.
• Mathematics, Computer Science
STOC '06
• 2006
One of the main open problems in the area of property-testing, which was raised in the 1996 paper of Goldreich, Goldwasser and Ron, is resolved by a purely combinatorial characterization of the graph properties that are testable with a constant number of queries.
• Mathematics, Computer Science
Proceedings 2001 IEEE International Conference on Cluster Computing
• 2001
Three theorems regarding testing graph properties in the adjacency matrix representation are presented and every graph property that can be tested making a number of queries that is independent of the size of the graph, can be so tested by uniformly selecting a set of vertices.
• Mathematics
SIAM J. Comput.
• 2007
It is shown that any property of bipartite graphs that is characterized by a finite collection of forbidden induced subgraphs is $\epsilon$-testable, with a number of queries that is polynomial in $1/\ep silon$.
• Mathematics, Computer Science
46th Annual IEEE Symposium on Foundations of Computer Science (FOCS'05)
• 2005
It is shown that a graph property P has an oblivious one-sided error tester, if and only if P is (semi) hereditary, and infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with one- sided error.
• Computer Science, Mathematics
STOC '05
• 2005
It is shown here that in the setting of the dense graph model, all testable properties are not only tolerantly testable, but also admit a constant query size algorithm that estimates the distance from the property up to any fixed additive constant.
• Mathematics
ArXiv
• 2011
It is shown that a weak partition with approximation parameter Epsilon may require as many as 2^{\Omega}(\epsilon^{-2}) parts, which is tight up to the implied constant and solves a problem studied by Lovász and Szegedy.
• Mathematics
STOC '06
• 2006
We define a distance of two graphs that reflects the closeness of both local and global properties. We also define convergence of a sequence of graphs, and show that a graph sequence is convergent if
• Mathematics
J. Comb. Theory, Ser. B
• 2017