AbstractLetGbe a 2‐edge connected graph with at least 5 vertices. For any given verticesa, b, c, anddinGwitha≠b, there exists inG3a hamiltonian path with endpointsaandbavoiding the edgecd, and there exists inG3∪{cd}a hamiltonian path with endpointsaandband containing the edgecd.Also, after removal of two edges or one edge and one vertex fromG3, the resulting graph is still hamilt