\(G\)-Reconstruction of Graphs

Let \(G\) be a group of permutations acting on an \(7\)-vertex set \(V\), and \(X\) and \(Y\) be two simple graphs on \(V\). We say that \(X\) and \(Y\) are \(G\)-isomorphic if \(Y\) belongs to the orbit of \(X\) under the action of \(G\). One can naturally generalize the reconstruction problems so that when \(G\) is \(S_v\), the symmetric group, we have the usual reconstruction problems. In this paper, we study \(G\)-edge reconstructibility of graphs. We prove some old and new results on edge reconstruction and reconstruction from end vertex deleted subgraphs.