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Local certification of forbidden subgraphs

Nicolas Bousquet, Linda Cook, Laurent Feuilloley, Théo Pierron, Sébastien Zeitoun

TL;DR

This paper starts the study of subgraph detection from the perspective of local certification, uncovering an interesting interplay between the optimal certificate size, the size of the forbidden subgraph, and the locality of the verification.

Abstract

Detecting specific structures in a network has been a very active theme of research in distributed computing for at least a decade. In this paper, we start the study of subgraph detection from the perspective of local certification. Remember that a local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. Our main question is: For a given graph $H$, what is the minimum certificate size that allows checking that the network does not contain $H$ as a (possibly induced) subgraph? We show a variety of lower and upper bounds, uncovering an interesting interplay between the optimal certificate size, the size of the forbidden subgraph, and the locality of the verification. Along the way we introduce several new technical tools, in particular what we call the \emph{layered map}, which is not specific to forbidden subgraphs and that we expect to be useful for certifying many other properties.

Local certification of forbidden subgraphs

TL;DR

This paper starts the study of subgraph detection from the perspective of local certification, uncovering an interesting interplay between the optimal certificate size, the size of the forbidden subgraph, and the locality of the verification.

Abstract

Detecting specific structures in a network has been a very active theme of research in distributed computing for at least a decade. In this paper, we start the study of subgraph detection from the perspective of local certification. Remember that a local certification is a distributed mechanism enabling the nodes of a network to check the correctness of the current configuration, thanks to small pieces of information called certificates. Our main question is: For a given graph , what is the minimum certificate size that allows checking that the network does not contain as a (possibly induced) subgraph? We show a variety of lower and upper bounds, uncovering an interesting interplay between the optimal certificate size, the size of the forbidden subgraph, and the locality of the verification. Along the way we introduce several new technical tools, in particular what we call the \emph{layered map}, which is not specific to forbidden subgraphs and that we expect to be useful for certifying many other properties.
Paper Structure (19 sections, 21 theorems, 1 figure, 1 table)

This paper contains 19 sections, 21 theorems, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Context
  3. Our results and techniques
  4. Additional related work and discussions
  5. Model and definitions
  6. Graph theory notions
  7. Local certification
  8. Lower bounds for paths (and extensions)
  9. Our lower bound construction
  10. Proof of Proposition \ref{['prop:common non-edge']}
  11. Upper bound core technique: layered maps
  12. Spread universal certification and graphs of large minimum degree
  13. Definitions: extended connected components, ECC table, and more
  14. Technical results
  15. Formal proofs of the forbidden subgraph certifications
  16. ...and 4 more sections

Key Result

Theorem 2

For $k>3$, the optimal certificate size for $K_k$-free graphs with verification radius $1$ is $\Theta(n)$. For $k=3$, it lies between $\Omega(n/e^{O(\sqrt{\log n})})$ and $O(n)$ bits.

Figures (1)

  • Figure 1: The graph $G_{k,n}(A,B)$. Each blob is a clique on $n$ vertices. The double thick edges represent the graphs $G_A$ and $G_B$. The other double edges represent the matchings. The double dashed edges represent the antimatchings.

Theorems & Definitions (25)

  • Theorem 2: BousquetEFZ24
  • Theorem 3
  • Conjecture 6
  • Theorem 7
  • Theorem 8
  • Theorem 9
  • Theorem 10
  • Theorem 11
  • Theorem 12
  • Theorem 12: BousquetEFZ24
  • ...and 15 more