A characterization of one-sided error testable graph properties in bounded degeneracy graphs

Abstract: We consider graph property testing in $p$-degenerate graphs under the random neighbor oracle model (Czumaj and Sohler, FOCS 2019). In this framework, a tester explores a graph by sampling uniform neighbors of vertices, and a property is testable with one-sided error if its query complexity is independent of the graph size. It is known that one-sided error testable properties for minor-closed families are exactly those that can be defined by forbidden subgraphs of bounded size. However, the much broader class of $p$-degenerate graphs allows for high-degree ``hubs" that can structurally hide forbidden subgraphs from local exploration. In this work, we provide a complete structural characterization of all properties testable with one-sided error in $p$-degenerate graphs. We show that testability is fundamentally determined by the connectivity of the forbidden structures: a property is testable if and only if its violations cannot be fragmented across disjoint high-degree neighborhoods. Our results define the exact structural boundary for testability under these constraints, accounting for both the connectivity of individual forbidden subgraphs and the collective behavior of the properties they define.

• The paper establishes a precise condition for when H-freeness is one-sided error testable, linking testability to the connectivity of subgraphs after removing independent sets.
• It introduces local exploration methods via constant-query testers that overcome challenges posed by high-degree hubs in p-degenerate graphs.
• The work unifies and extends prior results by providing a structural characterization that informs the design of efficient algorithms for sparse graph property testing.

A Structural Characterization of One-Sided Error Testability in Bounded Degeneracy Graphs

Introduction and Problem Formulation

This paper rigorously delineates the precise boundary for which monotone graph properties are testable with one-sided error in the random neighbor oracle model, when input graphs are from the class of $p$-degenerate graphs (i.e., graphs admitting an order of vertices so each has at most $p$ lower-indexed neighbors). Unlike the minor-closed case, bounded-degeneracy graphs possess unbounded degrees, allowing “hub” nodes that may occlude globally forbidden structures from local exploration algorithms. The random neighbor oracle model restricts exploration to locally available information: only existing edges can be observed, and negative information about non-edges is fundamentally inaccessible.

The main object of study is the testability, with query complexity dependent on parameters like degeneracy but not on the order of the graph, for arbitrary monotone properties—those closed under edge deletion. The paper focuses on one-sided error testing: the tester must accept every graph with the property with probability one and may reject only when it explicitly finds a subgraph violating the property.

The work generalizes and strictly subsumes the characterization previously established for proper minor-closed graph families, moving to degenerate graphs which can have arbitrarily high degrees and, crucially, can “hide” constant-size forbidden subgraphs within large neighborhoods, evading detection by local, constant-query oracles. This strict increase in expressiveness of the input family is shown to permit fundamentally different (non-testable) behavior for many graphs.

Main Theoretical Results

The authors’ structural characterization proceeds in two principal steps. First, the analysis reduces to forbidden subgraph properties: monotone properties are exactly those defined by the exclusion of a finite family $\mathcal{H}$ of constant-size forbidden subgraphs. The central technical challenge is to precisely characterize, for a given such forbidden family, when one-sided error testing is possible with constant query complexity (with respect to $n$).

The paper establishes a sharp structural dichotomy for when, for a fixed forbidden subgraph $H$, the property of being $H$-free is one-sided error testable in bounded degeneracy graphs:

$H$-freeness is one-sided error testable in bounded-degeneracy graphs if and only if for every independent set $S \subseteq V(H)$, the induced subgraph on $V(H) \setminus S$ is connected.

That is, the only independent sets that separate $H$ are those that—when removed—leave a disconnected graph. If such a “separating” independent set exists, $p$0-freeness is not testable with constant queries.

This result is constructive on both sides: for “hard” graphs (those possessing such obstacles), the authors explicitly construct infinite families of $p$1-degenerate graphs $p$2 that are $p$3-far from being $p$4-free, yet any constant-query one-sided error tester will fail with high probability to find a copy of $p$5 (see Figure 1). The construction leverages high-degree vertices (“hubs”) with “spoke” connections, distributing the support for the forbidden subgraph across the graph so that all necessary pieces are extremely unlikely to be simultaneously revealed by local sampling.

Figure 1: The graph $p$6 has a separation set $p$7.

This is made concrete with the canonical $p$8 lower bound: constructing a 2-degenerate graph with two hubs and two sets of paths between them—every $p$9 is split between these hubs and the path vertices, but no constant number of local explorations can recover a full cycle with non-negligible probability. The general construction (Figure 1) extends this to arbitrary obstacles.

For the case where no such separating independent set exists (i.e., for all independent sets, the remainder is connected), the authors design constant-query testers based on local explorations that, after appropriate “cleaning” (removing only edges affecting heavy-to-heavy connections), always discover a forbidden subgraph when the input is far from $\mathcal{H}$0-free.

Families of Forbidden Subgraphs and Collective Testability

For monotone properties defined by arbitrary families $\mathcal{H}$1 of forbidden subgraphs, the authors provide a full characterization: such a property is testable if and only if, for every non-testable $\mathcal{H}$2 (i.e., with a separating independent set $\mathcal{H}$3), and for each subset $\mathcal{H}$4 of cardinality $\mathcal{H}$5, there exists another $\mathcal{H}$6 in the family (a "sentinel") such that $\mathcal{H}$7 can be mapped into a “cactus” structure whose attachment points correspond to $\mathcal{H}$8, but in which each $\mathcal{H}$9-role is covered by at most one vertex.

The necessity is proved by adversarial constructions (“lower bound” graphs) which, unless some other forbidden subgraph detects the obstacle, demonstrate that no local search can succeed with constant probability. The sufficiency is shown via an inductive, algorithmic embedding of one of the sentinels, leveraging the structure of the “petals” (connected parts of cacti) to avoid collisions and ensure the probability mass is not lost at each branching.

Figure 2: The graph $n$0 has an obstacle set $n$1.

Figure 3: The graph $n$2 has a separation set $n$3.

Thus, the presence of testable "sentinel" forbidden subgraphs in the family functionally “breaks” the hard obstacle construction, a phenomenon exemplified by families such as $n$4 (the 4-cycle and the 10-star): although cycle-freeness is not testable in general, the presence of the forbidden star bounds the maximum degree, precluding the C4-obstacle and rendering the property testable.

Technical Approach and Canonical Testers

All results are formalized in the random neighbor oracle model, with explicit construction of canonical testers: procedures that start exploratory searches from $n$5 randomly chosen vertices, using bounded-depth BFS with random neighbor sampling, and only reject upon explicit discovery of a forbidden subgraph.

The lower bound arguments make rigorous use of birthday-paradox reasoning to quantify the vanishing probability of local testers finding all necessary pieces of a forbidden structure coordinated through high-degree hubs. For upper bounds, the analysis shows that in the absence of such “fragmented” structures, a small sample must, with high probability, suffice to witness a violation.

Implications and Future Directions

The characterization provided sharply distinguishes the random neighbor oracle testability landscape for $n$6-degenerate graphs, subsuming and generalizing earlier results for minor-closed families. The construction of the lower bound instances reveals the power of degree heterogeneity in enabling “hidden” forbidden structures, and the necessity of global connectivity among the pieces of a forbidden configuration for feasible detection.

The explicit characterization for families of forbidden subgraphs provides a usable framework for analyzing the testability of complex monotone properties in sparse graphs, with immediate implications for subgraph-freeness, hyperfiniteness, and structural graph parameters.

On a theoretical level, these results clarify the interplay between connectivity and local testability, suggesting further investigation into more refined models of local exploration, resource-bounded testers, and parameterized extensions (e.g., higher arboricity, local expansion constraints).

Conclusion

This work delivers a complete structural characterization of monotone properties one-sided error testable in the random neighbor oracle model for $n$7-degenerate graphs. The testability boundary is governed not merely by the presence of forbidden subgraphs, but by the fine-grained connectedness of these subgraphs and the combinatorics of how their “pieces” interact with high-degree neighborhoods. The results provide foundational insights into the granular mechanisms by which combinatorial structure constrains algorithmic property testing in sparse graph regimes, and the framework is broadly applicable for future explorations in local algorithms and combinatorial property testing.