The local k-neighborhood of a vertex v in an unweighted graph G=(V,E) with vertex set V and edge set E is the subgraph induced by all vertices of distance at most k from v. The rooted k-neighborhood of v is also called a k-disk around vertex v. If a graph has maximum degree bounded by a constant d, and k is also constant, the number of isomorphism classes of k-disks is constant as well. We can describe the local structure of a bounded-degree graph G by counting the number of isomorphic copies in G of each possible k-disk. We can summarize this information in form of a vector that has an entry for each isomorphism class of k-disks. The value of the entry is the number of isomorphic copies of the corresponding k-disk in G. We call this vector frequency vector of k-disks. If we only know this vector, what does it tell us about the structure of G? In this paper we will survey a series of papers in the area of Property Testing that leads to the following result (stated informally): There is a k=k(ε,d) such that for any planar graph G its local structure (described by the frequency vector of k-disks) determines G up to insertion and deletion of at most εd n edges (and relabelling of vertices). © 2014 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Sohler, C. (2014). What does the local structure of a planar graph tell us about its global structure? In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 8634 LNCS, pp. 44–49). Springer Verlag. https://doi.org/10.1007/978-3-662-44522-8_4
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