On Hamilton-Connectedness of Sierpiński Extension of Hamilton-Connected Graphs

Sierpiński networks have been extensively researched due to their fractal structure and their numerous scientific uses. The self-similar network, which is a generalized Sierpiński network, is generated by replicating the primary network. The extended Sierpiński networks are created by adding an additional vertex to a generic Sierpiński network and connecting it with the extreme vertices. A network is said to be Hamilton-connected if and only if there exists a Hamiltonian path connecting every two separate nodes. The problem of determining the Hamiltonicity of graphs exhibits NP-completeness. Determining if a graph is Hamiltonian, traceable, or Hamilton-connected is significant. In this present study, we investigate proving that the Sierpiński extension of any Hamilton-connected graph is Hamilton-connected. Also, we prove that generalized extended Sierpiński graphs S+(H,q) and S++(H,q) are Hamilton-connected when H is Hamilton-connected.