A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error

TL;DR

This paper introduces a sublinear, one-sided error property testing algorithm for outerplanarity and other forbidden minors in bounded-degree graphs, with query complexity significantly better than previous methods for certain graph classes.

Contribution

The paper presents a novel sublinear-time property testing algorithm for -minor freeness, including outerplanarity, with improved query complexity and structural analysis techniques.

Findings

Query complexity is O(n^{2/3}/\u03b5^5) for testing outerplanarity.

Either finds a small subgraph containing the forbidden minor or a large edge-cut inducing the minor.

Improves understanding of graph structure in property testing for forbidden minors.

Abstract

We consider one-sided error property testing of $\mathcal{F}$-minor freeness in bounded-degree graphs for any finite family of graphs $\mathcal{F}$ that contains a minor of $K_{2,k}$, the $k$-circus graph, or the $(k\times 2)$-grid for any $k\in\mathbb{N}$. This includes, for instance, testing whether a graph is outerplanar or a cactus graph. The query complexity of our algorithm in terms of the number of vertices in the graph, $n$, is $\tilde{O}(n^{2/3} / \epsilon^5)$. Czumaj et~al.\ showed that cycle-freeness and $C_k$-minor freeness can be tested with query complexity $\tilde{O}(\sqrt{n})$ by using random walks, and that testing $H$-minor freeness for any $H$ that contains a cycles requires $\Omega(\sqrt{n})$ queries. 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, which induces an $H$-minor.