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

Oded Lachish, Amit Levi, Ilan Newman, Felix Reidl

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.

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

Abstract

We consider graph property testing in -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 -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 -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.

Paper Structure

This paper contains 16 sections, 25 theorems, 12 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Our results
  3. Technical Overview
  4. Related work
  5. Preliminaries and Notations
  6. Testing $H$-freeness for $p$-degenerate graphs
  7. Testing $H$-freeness for 2-connected $H$
  8. Testing $\mathcal{H}$-freeness for a family $\mathcal{H}$ of forbidden subgraphs
  9. On $G$'s that are testable with respect to a hard to test $H_1$
  10. The testable cacti
  11. Phase 1: iterative pruning
  12. Phase 2: Probabilistic Cactus Embedding
  13. Testable $\{H_1,H\}$-freeness - the general case
  14. Extension to finite family of forbidden graphs
  15. Bounded-BFS simulation using random-oracle
  16. ...and 1 more sections

Key Result

Theorem 1

$H$-freeness is one-sided error testable for a $2$-connected graph $H$ if and only if for every independent set $S \subseteq V(H),$ the subgraph induced by $V(H)\setminus S$ is connected. $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: The graph $H$ has a separation set $S=\{a,b,c\}$. The graph $F$ has $m^2$ copies of $H$, one for every $(i,j) \in [m^2]$. Note that the vertex $c$ is present in each of the above copies.
  • Figure 2: The graph $H_1$ has an obstacle set $S=\{a,b\}$. The graph $H$ is a $4$-petal cactus, where petal $P_1$ is connected to petal $P_2$ with a vertex taking the role of $a$, $P_2$ is connected to $P_3$ with a vertex taking the role of $b$, $P_3$ is connected to $P_4$ via $a$.
  • Figure 3: The graph $H_1$ has a separation set $S=\{a,b,c\}$. The graph $H$ is a $5$-petal cactus, where petal $P_2$ is connected to petal $P_1$ with a vertex taking the role of $a$, to $P_3$ with a vertex taking the role of $b$, and to $P_4$ with $c$. Then, $P_4$ is connected to $P_5$ with a vertex with role $a$. Note that $P_1,$ which is a subgraph of the component $C_1$, could also be considered a subgraph of the component $C_2$. Thus, the structure of $H$ as a cactus is not unique. In particular, $H'$ is isomorphic to $H$, but its articulation points have different $S$-roles.
  • Figure 4: Simulation of bounded-depth BFS using random oracle.

Theorems & Definitions (58)

  • Theorem 1
  • Definition 1: 2-block
  • Definition 2: components
  • Definition 3: $p$-degenerate
  • Definition 4: $H$-appearance
  • Definition 5
  • Theorem 2
  • Lemma 2.2
  • proof
  • Definition 6: semi-bipartite structure
  • ...and 48 more