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Optimal local certification on graphs of bounded pathwidth

Dan Alden Baterisna, Yi-Jun Chang

TL;DR

The paper proves that any $oxed{ ext{$oxed{ extsf{MSO}_2}$}}$ property can be locally certified on graphs with bounded pathwidth using $oxed{O(\, obreakspace ext{log } n)}$-bit vertex labels. It introduces a novel $k$-lane recursive graph framework that enables $O(1)$-congestion embeddings of interval-based decompositions, allowing dynamic-programming-style MSO$_2$ certification to be implemented in a locally verifiable manner. This yields $O( obreakspace ext{log } n)$-bit proof labeling schemes for pathwidth-bounded graphs and, via the Excluding Forest Theorem, $O( obreakspace ext{log } n)$-bit schemes for $F$-minor-free graphs for any forest $F$, addressing an open question. The results are tight up to the known $oxed{ ext{Ω}( obreakspace ext{log } n)}$ lower bound and pave the way for extending optimal local certification to broader minor-closed graph classes. The techniques offer a structural route to certify MSO$_2$ properties in sparse graphs and may influence both theory and distributed applications in self-stabilizing and fault-tolerant networks.

Abstract

We present proof labeling schemes for graphs with bounded pathwidth that can decide any graph property expressible in monadic second-order (MSO) logic using $O(\log n)$-bit vertex labels. Examples of such properties include planarity, Hamiltonicity, $k$-colorability, $H$-minor-freeness, admitting a perfect matching, and having a vertex cover of a given size. Our proof labeling schemes improve upon a recent result by Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), which achieved the same result for graphs of bounded treewidth but required $O(\log^2 n)$-bit labels. Our improved label size $O(\log n)$ is optimal, as it is well-known that any proof labeling scheme that accepts paths and rejects cycles requires labels of size $Ω(\log n)$. Our result implies that graphs with pathwidth at most $k$ can be certified using $O(\log n)$-bit labels for any fixed constant $k$. Applying the Excluding Forest Theorem of Robertson and Seymour, we deduce that the class of $F$-minor-free graphs can be certified with $O(\log n)$-bit labels for any fixed forest $F$, thereby providing an affirmative answer to an open question posed by Bousquet, Feuilloley, and Pierron (Journal of Parallel and Distributed Computing 2024).

Optimal local certification on graphs of bounded pathwidth

TL;DR

The paper proves that any oxed{ extsf{MSO}_2} property can be locally certified on graphs with bounded pathwidth using -bit vertex labels. It introduces a novel -lane recursive graph framework that enables -congestion embeddings of interval-based decompositions, allowing dynamic-programming-style MSO certification to be implemented in a locally verifiable manner. This yields -bit proof labeling schemes for pathwidth-bounded graphs and, via the Excluding Forest Theorem, -bit schemes for -minor-free graphs for any forest , addressing an open question. The results are tight up to the known lower bound and pave the way for extending optimal local certification to broader minor-closed graph classes. The techniques offer a structural route to certify MSO properties in sparse graphs and may influence both theory and distributed applications in self-stabilizing and fault-tolerant networks.

Abstract

We present proof labeling schemes for graphs with bounded pathwidth that can decide any graph property expressible in monadic second-order (MSO) logic using -bit vertex labels. Examples of such properties include planarity, Hamiltonicity, -colorability, -minor-freeness, admitting a perfect matching, and having a vertex cover of a given size. Our proof labeling schemes improve upon a recent result by Fraigniaud, Montealegre, Rapaport, and Todinca (Algorithmica 2024), which achieved the same result for graphs of bounded treewidth but required -bit labels. Our improved label size is optimal, as it is well-known that any proof labeling scheme that accepts paths and rejects cycles requires labels of size . Our result implies that graphs with pathwidth at most can be certified using -bit labels for any fixed constant . Applying the Excluding Forest Theorem of Robertson and Seymour, we deduce that the class of -minor-free graphs can be certified with -bit labels for any fixed forest , thereby providing an affirmative answer to an open question posed by Bousquet, Feuilloley, and Pierron (Journal of Parallel and Distributed Computing 2024).

Paper Structure

This paper contains 62 sections, 15 theorems, 7 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Proof labeling schemes
  3. Complexity measure.
  4. Monadic second-order logic
  5. Algorithmic meta-theorems.
  6. Local certification.
  7. Minor-closed graph properties
  8. Our results
  9. Our contribution.
  10. Additional related work
  11. Preliminaries
  12. Basics tools in proof labeling schemes
  13. Edge certification.
  14. Graphs with designated vertices.
  15. MSO2 properties and k-terminal recursive graphs
  16. ...and 47 more sections

Key Result

Theorem 1

For any integer $k \geq 1$, for any $\mathsf{MSO}_2$ graph property $\phi$, there exists an $O(\log n)$-bit proof labeling scheme for the graph property $\phi \wedge (\operatorname{pathwidth}\leq k)$.

Figures (10)

  • Figure 1: Path decomposition and interval representation of a $6$-cycle.
  • Figure 2: Combining two $3$-terminal graphs into a $3$-terminal graph.
  • Figure 3: Weak completion and completion.
  • Figure 4: Graph terminology in \ref{['sect:embedding']}.
  • Figure 5: Case 1 in the proof of \ref{['low-completion-cost']}.
  • ...and 5 more figures

Theorems & Definitions (43)

  • Definition 1.1: Path decomposition ROBERTSON198339
  • Theorem 1
  • Corollary 1.2
  • proof
  • Proposition 2.1: feuilloley2023local
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4: borie1992automaticCOURCELLE199012
  • ...and 33 more