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

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.Comment: extended to testing outerplanarity, full version of ICALP pape

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

We consider one-sided error property testing of F-minor freeness in bounded-degree graphs for any finite family of graphs F that contains a minor of K_{2,k}, the k-circus graph, or the (k x 2)-grid for any k in 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 O~(n^{2/3} / epsilon^5). Czumaj et al. (2014) showed that cycle-freeness and C_k-minor freeness can be tested with query complexity 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

Testing Hamiltonicity (And Other Problems) in Minor-Free Graphs

In this paper we provide sub-linear algorithms for several fundamental problems in the setting in which the input graph excludes a fixed minor, i.e., is a minor-free graph. In particular, we provide the following algorithms for minor-free unbounded degree graphs. 1) A tester for Hamiltonicity with two-sided error with poly(1/?)-query complexity, where ? is the proximity parameter. 2) A local algorithm, as defined by Rubinfeld et al. (ICS 2011), for constructing a spanning subgraph with almost minimum weight, specifically, at most a factor (1+?) of the optimum, with poly(1/?)-query complexity. Both our algorithms use partition oracles, a tool introduced by Hassidim et al. (FOCS 2009), which are oracles that provide access to a partition of the graph such that the number of cut-edges is small and each part of the partition is small. The polynomial dependence in 1/? of our algorithms is achieved by combining the recent poly(d/?)-query partition oracle of Kumar-Seshadhri-Stolman (ECCC 2021) for minor-free graphs with degree bounded by d. For bounded degree minor-free graphs we introduce the notion of covering partition oracles which is a relaxed version of partition oracles and design a poly(d/?)-time covering partition oracle for this family of graphs. Using our covering partition oracle we provide the same results as above (except that the tester for Hamiltonicity has one-sided error) for minor-free bounded degree graphs, as well as showing that any property which is monotone and additive (e.g. bipartiteness) can be tested in minor-free graphs by making poly(d/?)-queries. The benefit of using the covering partition oracle rather than the partition oracle in our algorithms is its simplicity and an improved polynomial dependence in 1/? in the obtained query complexity

Every testable (infinite) property of bounded-degree graphs contains an infinite hyperfinite subproperty

One of the most fundamental questions in graph property testing is to characterize the combinatorial structure of properties that are testable with a constant number of queries. We work towards an answer to this question for the bounded-degree graph model introduced in [GR02], where the input graphs have maximum degree bounded by a constant d. In this model, it is known (among other results) that every hyperfinite property is constant-query testable [NS13], where, informally, a graph property is hyperfinite, if for every δ > 0 every graph in the property can be partitioned into small connected components by removing δn edges. In this paper we show that hyperfiniteness plays a role in every testable property, i.e. we show that every testable property is either finite (which trivially implies hyperfiniteness and testability) or contains an infinite hyperfinite subproperty. A simple consequence of our result is that no infinite graph property that only consists of expander graphs is constant-query testable. Based on the above findings, one could ask if every infinite testable non-hyperfinite property might contain an infinite family of expander (or near-expander) graphs. We show that this is not true. Motivated by our counterexample we develop a theorem that shows that we can partition the set of vertices of every bounded degree graph into a constant number of subsets and a separator set, such that the separator set is small and the distribution of k-discs on every subset of a partition class, is roughly the same as that of the partition class if the subset has small expansion