A Characterization of Graph Properties Testable for General Planar Graphs with one-Sided Error (It's all About Forbidden Subgraphs)

Abstract

The problem of characterizing testable graph properties (properties that can be tested with a number of queries independent of the input size) is a fundamental problem in the area of property testing. While there has been some extensive prior research characterizing testable graph properties in the dense graphs model and we have good understanding of the bounded degree graphs model, no similar characterization has been known for general graphs, with no degree bounds. In this paper we take on this major challenge and consider the problem of characterizing all testable graph properties in general planar graphs.We consider the model in which a general planar graph can be accessed by the random neighbor oracle that allows access to any given vertex and access to a random neighbor of a given vertex. We show that, informally, a graph property P is testable with one-sided error for general planar graphs if and only if testing P can be reduced to testing for a finite family of finite forbidden subgraphs. While our presentation focuses on planar graphs, our approach extends easily to general minor-free graphs.Our analysis of the necessary condition relies on a recent construction of canonical testers in the random neighbor oracle model that is applied here to the one-sided error model for testing in planar graphs. The sufficient condition in the characterization reduces the problem to the task of testing Hfreeness in planar graphs, and is the main and most challenging technical contribution of the paper: we show that for planar graphs (with arbitrary degrees), the property of being H-free is testable with one-sided error for every finite graph H, in the random neighbor oracle model.

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“…Then, in Sections III-VII, we present the main technical contribution of this paper, a complete analysis showing the sufficient part of our characterization of testable properties in planar graphs, that for any finite graph H, testing H-freeness is testable in planar graphs. The analysis here is split into several sections, with some auxiliary and technical results deferred to the full version of the paper [13]. We begin in Section III with an outline of the proof of testing H-freeness, focusing on connected H. Then, in Section IV, we present our tester and define our framework.…”

“…While Theorem 11 from [9] allows to simplify the analysis of testable properties, the analysis as in Theorem 12 obtains non-uniform testers, in the sense of the dependency on n. We could make our result uniform by considering a special class of uniform testers, which we call oblivious testers, that capture the essence of testers of testable properties in the flavor of Theorem 11 (see [6] for a similar notion in the context of testing dense graphs). This characterization is discussed in details in the full version of the paper [13].…”

“…We first describe our algorithm for testing Hfreeness of planar graphs with arbitrary degrees and provide the high level structure of its analysis. We defer most of technical details to Sections IV-VII and the full version of the paper [13].…”

“…(In particular, informally, in our analysis we would like to argue that there is always a constant fraction of low-degree vertices.) While we do not have a useful characterization of planar hypergraphs, we will be able to model some planarity-like properties of the hypergraphs using some special graph reduction (via shadow graphs), see Lemma 29 (and for more details, see the full version of the paper [13]).…”

“…2) Auxiliary tools: Simplifying condition (a) of Lemma 17: (via edge-disjoint copies of H): We show that one can simplify condition (a) of Lemma 17 for the special case when the subgraph G of G is a union of a linear number of edgedisjoined colored copies of H (a similar approach has been also used in [11]). That is, if there is a graph G[Q] with a linear number of edge-disjoint colored copies of H, then Lemma 18 shows that there is always a subset Q ⊆ Q with cardinality The proof of Lemma 18, as a natural extension of the approach from [11], is deferred to the full version of the paper [13].…”