The Reconstruction Number of a Lexicographic Sum of Cliques

Abstract

The clique sum $\mathrm{\Sigma }=G[G_1,G_2,\dots ,G_n]$ is the lexicographic sum over $G$ where each fiber $G_i$ is a clique. We show the reconstruction number of $\mathrm{\Sigma }$ is three unless $\mathrm{\Sigma }$ is vertex transitive and $G$ has order at least two. In the latter case, it follows that $\mathrm{\Sigma }=G[K_m]$ 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.