Linear-time certifying recognition algorithms and forbidden induced subgraphs

We give the first linear-time certifying algorithms to recognize split graphs, threshold graphs, chain graphs, co-chain graphs and trivially perfect graphs, with sublinear certificates for negative output. In case of membership in a given graph class our algorithms provide as certificate a structure for the input graph that characterizes the class, and in case of non-membership they provide as certificate a forbidden induced subgraph of the class. The certificates of membership can be authenticated in time O(n + m) and the certificates of non-membership can be authenticated in time O(1).