## Three Theorems regarding Testing Graph Properties

#### Abstract

Property testing is a relaxation of decision problems in which it is required to distinguish yes-instances (i.e., objects having a predetermined property) from instances that are far from any yes-instance. We presents three theorems regarding testing graph properties in the adjacency matrix representation. More specifically, these theorems relate to the project of characterizing graph properties according to the complexity of testing them (in the adjacency matrix representation).

The first theorem is that there exist monotone graph properties in NP for which testing is very hard (i.e., requires to examine a constant fraction of the entries in the matrix). Our second theorem is that 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 and accepting iff the induced subgraph has some fixed graph property (which is not necessarily the same as the one being tested). Our third theorem refers to the framework of graph partition problems, and is a characterization of the subclass of properties that can be tested using a one-sided error tester making a number of queries that is independent of the size of the graph.

#### Material available on-line

• The first posted version, 2001.
• Comments regarding the first posted version, 2002.
• Errata to the final/journal version (re the 2nd Theorem), 2005. [PDF versions]
• Another errata, this one minor and referring to the 1st theorem.
Due to switching between proofs, we forgot to adapt some constants in the proof. Specifically, the threshold for light weigth strings should be modified from relative Hamming weight of $1/3$ to, say, relative Hamming weight of $0.49$. This allows to prove Claim 3.1 as stated, where the problem is upper bounding the number of graphs that are $0.1$-close to some final graph. This modification allows to upperbound the number of final graphs by $$2^{2t+o(t)}\cdot(N!)\cdot2^{0.51 {N\choose 2}} = 2^{(0.02 + o(1) + 0.51)\cdot {N\choose 2}}$$ (rather than $2^{0.7 {N\choose 2}}$ as in Eq. (1))., Each such graph is $0.1$-close to at most $2^{0.469 {N\choose 2}}$ graphs, and so the total number of graphs that are $0.1$-close to some final graph is at most $2^{(0.999+o(1))\cdot {N\choose 2}} =o(2^{{N\choose 2}})$.
[Sept 2008]