## Abstract

A D-disc around a vertex v of a graph G = (V,E) is the subgraph induced by all vertices of distance at most D from v. We show that the structure of an outerplanar graph on n vertices is determined, up to modification (insertion or deletion) of at most ϵn edges, by a set of D-discs around the vertices, for D = D(ϵ) that is independent of the size of the graph. Such a result was already known for planar graphs (and any hyperfinite graph class), in the limited case of bounded degree graphs (that is, their maximum degree is bounded by some fixed constant, independent of |V |). We prove this result with no assumption on the degree of the graph. A pure combinatorial consequence of this result is that two outerplanar graphs that share the same local views are close to be isomorphic. We also obtain the following property testing results in the sparse graph model: graph isomorphism is testable for outerplanar graphs by poly(log n) queries. every graph property is testable for outerplanar graphs by poly(log n) queries. We note that we can replace outerplanar graphs by a slightly more general family of k-edgeouterplanar graphs. The only previous general testing results, as above, where known for forests (Kusumoto and Yoshida), and for some power-law graphs that are extremely close to be bounded degree hyperfinite (by Ito).

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 19th International Workshop, APPROX 2016 and 20th International Workshop, RANDOM 2016 |

Editors | Klaus Jansen, Claire Mathieu, Jose D. P. Rolim, Chris Umans |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770187 |

DOIs | |

State | Published - 1 Sep 2016 |

Event | 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016 - Paris, France Duration: 7 Sep 2016 → 9 Sep 2016 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 60 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2016 and the 20th International Workshop on Randomization and Computation, RANDOM 2016 |
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Country/Territory | France |

City | Paris |

Period | 7/09/16 → 9/09/16 |

## Keywords

- Isomorphism
- Outerplanar graphs
- Property testing

## ASJC Scopus subject areas

- Software