Lexicographic products with high reconstruction numbers

The reconstruction number of a graph is the smallest number of vertex-deleted subgraphs needed to uniquely determine the graph up to isomorphism. Bollobás showed that almost all graphs have reconstruction number equal to three. McMullen and Radziszowski published a catalogue of all graphs on at most ten vertices with reconstruction number greater than three. We introduce constructions that generalize the examples identified in their work. In particular, we use lexicographic products of vertex transitive graphs with certain starter graphs from the work of Myrvold and from the work of Harary and Plantholt to generate new infinite families of graphs with high reconstruction numbers. In the process, we settle a question of McMullen and Radziszowski.