Local certification of MSO properties for bounded treedepth graphs

The graph model checking problem consists in testing whether an input graph satis es a given logical formula. In this paper, we study this problem in a distributed setting, namely local certi cation. The goal is to assign labels to the nodes of a network to certify that some given property is satis ed, in such a way that the labels can be checked locally. We rst investigate which properties can be locally certi ed with small certi cates. Not surprisingly, this is almost never the case, except for not very expressive logic fragments. Following the steps of Courcelle-Grohe, we then look for meta-theorems explaining what happens when we parameterize the problem by some standard measures of how simple the graph classes are. In that direction, our main result states that any MSO formula can be locally certi ed on graphs with bounded treedepth with a logarithmic number of bits per node, which is the golden standard in certi cation. 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms → Distributed algorithms