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A subquadratic certification scheme for P5-free graphs

Nicolas Bousquet, Sébastien Zeitoun

TL;DR

This work presents the first subquadratic local certification scheme for the property of $P_5$-freeness in graphs, achieving certificates of size $O(n^{3/2})$ under a radius-1 verification model. The authors leverage a structural theorem (dominating set as a clique or an induced $P_3$) to build valid tree partitions of the graph, and design a three-part certificate scheme—$ ext{Neighbors}$, $ ext{TreePartitioning}$, and $ ext{Pieces}$—that enables local verification to detect induced $P_5$s. The approach sidesteps previous radius-2 techniques by embedding information about the partition and vertex neighborhoods into compact certificates, with a careful encoding that guarantees both completeness and soundness. This result advances the understanding of how structural graph properties can be certified with subquadratic resources and suggests directions for further tightening bounds for longer forbidden paths.

Abstract

In local certification, vertices of a $n$-vertex graph perform a local verification to check if a given property is satisfied by the graph. This verification is performed thanks to certificates, which are pieces of information that are given to the vertices. In this work, we focus on the local certification of $P_5$-freeness, and we prove a $O(n^{3/2})$ upper bound on the size of the certificates, which is (to our knowledge) the first subquadratic upper bound for this property.

A subquadratic certification scheme for P5-free graphs

TL;DR

This work presents the first subquadratic local certification scheme for the property of -freeness in graphs, achieving certificates of size under a radius-1 verification model. The authors leverage a structural theorem (dominating set as a clique or an induced ) to build valid tree partitions of the graph, and design a three-part certificate scheme—, , and —that enables local verification to detect induced s. The approach sidesteps previous radius-2 techniques by embedding information about the partition and vertex neighborhoods into compact certificates, with a careful encoding that guarantees both completeness and soundness. This result advances the understanding of how structural graph properties can be certified with subquadratic resources and suggests directions for further tightening bounds for longer forbidden paths.

Abstract

In local certification, vertices of a -vertex graph perform a local verification to check if a given property is satisfied by the graph. This verification is performed thanks to certificates, which are pieces of information that are given to the vertices. In this work, we focus on the local certification of -freeness, and we prove a upper bound on the size of the certificates, which is (to our knowledge) the first subquadratic upper bound for this property.

Paper Structure

This paper contains 6 sections, 7 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Model and definitions
  3. Graphs
  4. Local certification
  5. Valid tree partitions
  6. Proof of Theorem \ref{['thm:p5_free']}.

Key Result

Theorem 1

The optimal size of the certificates for $K_k$-freness is $\Theta(n)$ if $k\geqslant 4$, and lies between $\Omega(n/e^{O(\sqrt{\log n})})$ and $O(n)$ if $k=3$.

Figures (2)

  • Figure 1: A rooted tree whose encoding is 0011010001011011.
  • Figure 2: The edges and non-edges seen by $v_3$, $v_2$ and $v_4$ in $P$, using just the part of $\mathsf{Neighbors}$ of the certificates. The edges are represented by the normal edges, the non-edges are represented by the dotted ones, and the red dashed edges represent the unknown edges or non-edges.

Theorems & Definitions (9)

  • Theorem 1: BousquetEFZ24, DruckerKO13
  • Theorem 2: BousquetCFPZ24+
  • Theorem 3: BousquetCFPZ24+
  • Conjecture 4: BousquetCFPZ24+
  • Theorem 5
  • Theorem 6: bacso1990dominating
  • Definition 7
  • Lemma 8
  • Theorem 8