 
Summary: Every Monotone Graph Property is Testable
Noga Alon
Asaf Shapira
Abstract
A graph property is called monotone if it is closed under removal of edges and vertices. Many
monotone graph properties are some of the most wellstudied properties in graph theory, and the
abstract family of all monotone graph properties was also extensively studied. Our main result in
this paper is that any monotone graph property can be tested with onesided error, and with query
complexity depending only on . This result unifies several previous results in the area of property
testing, and also implies the testability of wellstudied graph properties that were previously not
known to be testable. At the heart of the proof is an application of a variant of Szemer´edi's
Regularity Lemma. The main ideas behind this application may be useful in characterizing all
testable graph properties, and in generally studying graph property testing.
As a byproduct of our techniques we also obtain additional results in graph theory and property
testing, which are of independent interest. One of these results is that the query complexity of
testing testable graph properties with onesided error may be arbitrarily large. Another result,
which significantly extends previous results in extremal graphtheory, is that for any monotone
graph property P, any graph that is far from satisfying P, contains a subgraph of size depending
on only, which does not satisfy P. Finally, we prove the following compactness statement: If a
graph G is far from satisfying a (possibly infinite) set of monotone graph properties P, then it
