Summary: Homomorphisms in Graph Property Testing - A Survey
Dedicated to Jaroslav Nesetril on the occasion of his 60th
Property-testers 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 property-testers 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 property-testing.
1.1 Property-testing 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.