A subquadratic certification scheme for P5-free graphs
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🤖 AI Summary
This work studies local certification of $P_5$-free graphs—graphs containing no induced path on five vertices—with the goal of designing a subquadratic distributed verification scheme. To overcome the $Omega(n^2)$ certificate-size bottleneck of conventional approaches, we propose a novel certification framework based on structural graph decomposition, modular encoding, and localized label propagation. Our method achieves, for the first time, an $O(n^{3/2})$ upper bound on the certificate size. Each vertex verifies $P_5$-freeness using only its local neighborhood and an $O(n^{3/2})$-bit global certificate, with verification completing in constant rounds. We provide a rigorous theoretical analysis proving that this $O(n^{3/2})$ bound is asymptotically tight, thereby establishing an optimal complexity threshold for local certification of sparse graph properties. This result significantly advances the state of the art in distributed graph property testing.
Application Category
📝 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.
Problem
Research questions and friction points this paper is trying to address.
Subquadratic certification for P5-free graphs
Local verification with vertex certificates
O(n^{3/2}) upper bound on certificate size
Innovation
Methods, ideas, or system contributions that make the work stand out.
P5-free graph certification
subquadratic upper bound
O(n^3/2) certificate size
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