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Local certification of MSO properties for bounded treedepth graphs [article]

Nicolas Bousquet, Laurent Feuilloley, Théo Pierron
2021 arXiv   pre-print
In that direction, our main result states that any MSO formula can be locally certified on graphs with bounded treedepth with a logarithmic number of bits per node, which is the golden standard in certification  ...  We first investigate which properties can be locally certified with small certificates. Not surprisingly, this is almost never the case, except for not very expressive logic fragments.  ...  The next problem in line is then MSO-model checking for graphs of bounded treedepth.  ... 
arXiv:2110.01936v1 fatcat:344p5alkczaxfgn7xnmm63ah2q

What Can Be Certified Compactly? Compact local certification of MSO properties in tree-like graphs

Laurent Feuilloley, Nicolas Bousquet, Théo Pierron
2022 Proceedings of the 2022 ACM Symposium on Principles of Distributed Computing  
We first prove that an optimal certification for bounded treedepth uses certificates  ...  We then move on to graphs of bounded treedepth, a well-known parameter that basically measures how far a graph is from a star.  ...  ACKNOWLEDGMENTS We thank Louis Esperet for fruitful discussions, Rotem Oshman for pointing out references about triangle-free graphs certification, Anca Muscholl for discussions about tree automata.  ... 
doi:10.1145/3519270.3538416 fatcat:4z2ials4ofhpblutphqq3o7mdm

What can be certified compactly? [article]

Nicolas Bousquet and Laurent Feuilloley and Théo Pierron
2022 arXiv   pre-print
Recently, a series of papers have proved that several well-known network properties have compact local certifications: planarity, bounded-genus, etc.  ...  We take a similar approach and prove the first meta-theorems for local certification. (See the abstract of the pdf for more details.)  ...  INTRODUCTION 1.Local certification In this work, we are interested in the locality of graph properties. For example, consider the property "the graph has maximum degree three".  ... 
arXiv:2202.06065v1 fatcat:6kg5qprkl5ai3dnpppehl6fdqe

Extended MSO Model Checking via Small Vertex Integrity [article]

Tatsuya Gima, Yota Otachi
2022 arXiv   pre-print
We study the model checking problem of an extended 𝖬𝖲𝖮 with local and global cardinality constraints, called 𝖬𝖲𝖮^𝖦𝖫_𝖫𝗂𝗇, introduced recently by Knop, Koutecký, Masařík, and Toufar [Log.  ...  We show that the problem is fixed-parameter tractable parameterized by vertex integrity, where vertex integrity is a graph parameter standing between vertex cover number and treedepth.  ...  Acknowledgements The authors thank Michael Lampis and Valia Mitsou for fruitful discussions and sharing a preliminary version of [45] .  ... 
arXiv:2202.08445v3 fatcat:whndbaukenasjjdprpfqlwsc5q

A Meta-Theorem for Distributed Certification [article]

Pierre Fraigniaud, Pedro Montealegre, Ivan Rapaport, Ioan Todinca
2021 arXiv   pre-print
Specifically, we prove that, for every boolean predicate on graphs definable in the monadic second-order (MSO) logic of graphs, there exists a distributed certification mechanism using certificates on  ...  O(log^2n) bits in n-node graphs of bounded treewidth, with a verification protocol involving a single round of communication between neighbors.  ...  The authors are thankful to Eric Remila for fruitful discussions on certification schemes related to the one considered in this paper.  ... 
arXiv:2112.03195v1 fatcat:vslmebivsbadxim52dsmwnkieq

Parameterized Complexity of (A,ℓ)-Path Packing [article]

Rémy Belmonte, Tesshu Hanaka, Masaaki Kanzaki, Masashi Kiyomi, Yasuaki Kobayashi, Yusuke Kobayashi, Michael Lampis, Hirotaka Ono, Yota Otachi
2020 arXiv   pre-print
Given a graph G = (V,E), A ⊆ V, and integers k and ℓ, the (A,ℓ)-Path Packing problem asks to find k vertex-disjoint paths of length ℓ that have endpoints in A and internal points in V ∖ A.  ...  Among other results, we show that the problem is polynomial-time solvable when ℓ< 3, while it is NP-complete for constant ℓ> 4.  ...  treedepth + because the maximum length of a path in a graph is bounded by a function of treedepth [27, Section 6.2].  ... 
arXiv:2008.03448v1 fatcat:qqhkwar3xjex3hvckwytlduske