On Testing Minor-Freeness in Bounded Degree Graphs With One-Sided Error

We present a one-sided error property testing algorithm for H-minor freeness in boundeddegree graphs for any minor H that is a minor of the (k× 2)-grid (for any k ∈ N). This includes, for example, testing whether a graph is a cactus graph and testing minor-freeness for minors which are cycles with parallel chords. The query complexity of our algorithm in terms of the number of vertices in the graph, n, is Õ(n/ ). Czumaj et al. showed that cycle-freeness and Ck-minor freeness can be tested with query complexity Õ( √ n). In contrast to these results, we analyze the structure of the graph and show that either we can find a subgraph of sublinear size that includes the forbidden minor, H, or we can find a pair of disjoint subsets of vertices whose edge-cut is large, and such that the induced subgraph of each subset is connected. We then prove a combinatorial lemma which shows that the latter structure includes H as a minor. ∗The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n◦ 307696. †TU Dortmund, Dortmund, Germany, [email protected] ‡MPI for Informatics, Saarbrücken 66123, Germany, [email protected] §TU Dortmund, Dortmund, Germany, [email protected] ¶Universitat Politècnica de Catalunya, Barcelona, Spain, [email protected] ar X iv :1 70 7. 06 12 6v 2 [ cs .D S] 2 0 Ju l 2 01 7