 
Summary: Homomorphisms in Graph Property Testing  A Survey
Dedicated to Jaroslav Nesetril on the occasion of his 60th
birthday
Noga Alon
Asaf Shapira
Abstract
Propertytesters are fast randomized algorithms for distinguishing between graphs (and other
combinatorial structures) satisfying a certain property, from those that are far from satisfying it.
In many cases one can design propertytesters whose running time is in fact independent of the
size of the input. In this paper we survey some recent results on testing graph properties. A
common thread in all the results surveyed is that they rely heavily on the simple yet useful notion
of graph homomorphism. We mainly focus on the combinatorial aspects of propertytesting.
1 Introduction
1.1 Propertytesting background
The meta problem in the area of property testing is the following: Design a randomized algorithm,
which given a combinatorial structure S, can distinguish with high probability between the case that
S satisfies some property P from the case that S is far from satisfying P. Here S is said to be
far from satisfying P if an fraction of its representation should be modified in order to make S
satisfy P. The main goal is to design randomized algorithms, which look at a very small portion of
the input, and using this information distinguish with high probability between the above two cases.
