On Reconstructing Graphs from \(n -2\) Cards

Abstract

Let \( G \) and \( H \) be graphs on \( n+2 \) vertices \( \{u_1, u_2, \ldots, u_n, x, y\} \) such that \( G – u_i \cong H – u_i \), for \( i = 1, 2, \ldots, n \). Recently, Ramachandran, Monikandan, and Balakumar have shown in a sequence of two papers that if \( n \geq 9 \), then \( |\varepsilon(H) – \varepsilon(G)| \leq 1 \). In this paper, we present a simpler proof of their theorem, using a counting lemma.