The 2-steps Hamiltonian Subdivision Graphs of Cycles with a Chord

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A \((p, q)\)-graph \(G = (V, E)\) is said to be \(AL(k)\)-traversal if there exists a sequence of vertices \(\{v_1, v_2, \ldots, v_p\}\) such that for each \(i = 1, 2, \ldots, p-1\), the distance between \(v_i\) and \(v_{i+1}\) is equal to \(k\). We call a graph \(G\) a \(k\)-steps Hamiltonian graph if it has an \(AL(k)\)-traversal in \(G\) and the distance between \(v_p\) and \(v_1\) is \(k\). In this paper, we completely classify whether a subdivision graph of a cycle with a chord is \(2\)-steps Hamiltonian.