The Reconstruction Number of a Lexicographic Sum of Cliques

The clique sum \(\Sigma = G[G_1,G_2,\ldots,G_n]\) is the lexicographic sum over \(G\) where each fiber \(G_i\) is a clique. We show the reconstruction number of \(\Sigma\) is three unless \(\Sigma\) is vertex transitive and \(G\) has order at least two. In the latter case, it follows that \(\Sigma = G[K_m]\) is a lexicographic product, and the reconstruction number is \(m+2\). This complements the bounds of Brewster, Hahn, Lamont, and Lipka. It also extends the work of Myrvold and Molina.