Hamiltonian Complete Number of Some Variants of Caterpillar Graphs

A graph $G$ is said to be Hamiltonian if it contains a spanning cycle. In this work, we investigate the Hamiltonian completeness of certain classes of caterpillar graphs, which are trees with a central path to which all other vertices are adjacent. For a non-Hamiltonian graph $G$, the Hamiltonian complete number $\lambda_H(G)$ is the minimum number of edges that must be added to $G$ to make it Hamiltonian. We focus on both regular and irregular caterpillar graphs, deriving explicit formulas for $\lambda_H(G)$ in various cases. Specifically, we show that for a regular caterpillar graph $G_{n(k)}$ where each vertex on the central path is adjacent to $k$ leaves, $\lambda_H(G_{n(k)}) = n(k-1)$. We also explore irregular caterpillar graphs, where the number of leaves adjacent to each vertex on the central path varies, and provide bounds for $\lambda_H(G)$ in these cases. Our results contribute to the understanding of Hamiltonian properties in tree-like structures and have potential applications in network design and optimization.