Linear-time certifying algorithms for recognizing split graphs and related graph classes∗

We give the first linear-time certifying algorithms to recognize split graphs, threshold graphs, bipartite chain graphs, cobipartite 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(n).