Non-Deterministic Graph Property Testing

A property of finite graphs is called non-deterministically testable if it has a ‘certificate’ such that once the certificate is specified, its correctness can be verified by random local testing. In this paper we study certificates that consist of one or more unary and/or binary relations on the nodes, in the case of dense graphs. Using the theory of graph limits, we prove that non-deterministically testable properties are also deterministically testable.

Download Full-text

Flows on measurable spaces

Geometric and Functional Analysis ◽

10.1007/s00039-021-00561-9 ◽

2021 ◽

Author(s):

László Lovász

Keyword(s):

Directed Graphs ◽

Flow Theory ◽

Space Structure ◽

Borel Space ◽

Bounded Degree ◽

Counting Measure ◽

Finite Graphs ◽

Graph Limits ◽

Dense Graphs ◽

Bounded Degree Graphs

AbstractThe theory of graph limits is only understood to a somewhat satisfactory degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that one of the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with a stationary distribution). This motivates our goal to extend some important theorems from finite graphs to Markov spaces or, more generally, to measurable spaces. In this paper, we show that much of flow theory, one of the most important areas in graph theory, can be extended to measurable spaces. Surprisingly, even the Markov space structure is not fully needed to get these results: all we need a standard Borel space with a measure on its square (generalizing the finite node set and the counting measure on the edge set). Our results may be considered as extensions of flow theory for directed graphs to the measurable case.

Download Full-text

Approximating Cayley Diagrams Versus Cayley Graphs

Combinatorics Probability Computing ◽

10.1017/s0963548311000733 ◽

2012 ◽

Vol 21 (4) ◽

pp. 635-641

Author(s):

ÁDÁM TIMÁR

Keyword(s):

Cayley Graph ◽

Hamiltonian Cycle ◽

Cayley Graphs ◽

The Other ◽

Finite Graphs ◽

Similar Construction ◽

Graph Property

We construct a sequence of finite graphs that weakly converge to a Cayley graph, but there is no labelling of the edges that would converge to the corresponding Cayley diagram. A similar construction is used to give graph sequences that converge to the same limit, and such that a Hamiltonian cycle in one of them has a limit that is not approximable by any subgraph of the other. We give an example where this holds, but convergence is meant in a stronger sense. This is related to whether having a Hamiltonian cycle is a testable graph property.

Download Full-text

On the Benefits of Adaptivity in Property Testing of Dense Graphs

Algorithmica ◽

10.1007/s00453-008-9237-4 ◽

2008 ◽

Vol 58 (4) ◽

pp. 811-830 ◽

Cited By ~ 3

Author(s):

Mira Gonen ◽

Dana Ron

Keyword(s):

Property Testing ◽

Dense Graphs

Download Full-text

On the Typical Structure of Graphs in a Monotone Property

The Electronic Journal of Combinatorics ◽

10.37236/4266 ◽

2014 ◽

Vol 21 (3) ◽

Cited By ~ 1

Author(s):

Svante Janson ◽

Andrew J. Uzzell

Keyword(s):

Typical Structure ◽

Monotone Property ◽

Graph Limits ◽

Graph Property ◽

Colorable Graph

Given a graph property $\mathcal{P}$, it is interesting to determine the typical structure of graphs that satisfy $\mathcal{P}$. In this paper, we consider monotone properties, that is, properties that are closed under taking subgraphs. Using results from the theory of graph limits, we show that if $\mathcal{P}$ is a monotone property and $r$ is the largest integer for which every $r$-colorable graph satisfies $\mathcal{P}$, then almost every graph with $\mathcal{P}$ is close to being a balanced $r$-partite graph.

Download Full-text