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The graph model checking problem consists in testing whether an input graphsatisfies a given logical formula. In this paper, we study this problem in adistributed setting, namely local certification. The goal is to assign labelsto the nodes of a network to certify that some given property is satisfied, insuch a way that the labels can be checked locally.  We first investigate which properties can be locally certified with smallcertificates. Not surprisingly, this is almost never the case, except for notvery expressive logic fragments. Following the steps of Courcelle-Grohe, wethen look for meta-theorems explaining what happens when we parameterize theproblem by some standard measures of how simple the graph classes are. In thatdirection, our main result states that any MSO formula can be locally certifiedon graphs with bounded treedepth with a logarithmic number of bits per node,which is the golden standard in certification.

Local certification of MSO properties for bounded treedepth graphs