Double Shells with Two Pendant Edges at the Apex are \(k\)-graceful

A double shell is defined to be two edge-disjoint shells with a common apex. In this paper, we prove that double shells (where the shell orders are \(m\) and \(2m+1\)) with exactly two pendant edges at the apex are \(k\)-graceful when \(k=2\). We extend this result to double shells of any order \(m\) and \(\ell\) (where \(m \geq 3\) and \(\ell \geq 3\)) with exactly two pendant edges at the apex.