Atoll decompositions of graphs

An island decomposition of a graph G consists of a set of vertex-disjoint paths which cover the vertex set of G. If the endpoints of the paths are mutually nonadjacent, then we have an atoll decomposition. We characterize graphs requiring two paths in an island decomposition yet having no atoll decomposition. Results are given relating atoll decompositions to cutpoints and Hamiltonian blocks.