Editing Outerplanar graph (section)

===Biconnectivity and Hamiltonicity===
An outerplanar graph is [[biconnected graph|biconnected]] if and only if the outer face of the graph forms a [[cycle (graph theory)|simple cycle]] without repeated vertices. An outerplanar graph is [[Hamiltonian cycle|Hamiltonian]] if and only if it is biconnected; in this case, the outer face forms the unique Hamiltonian cycle.<ref>{{harvtxt|Chartrand|Harary|1967}}; {{harvtxt|Sysło|1979}}.</ref> More generally, the size of the longest cycle in an outerplanar graph is the same as the number of vertices in its largest [[biconnected component]]. For this reason finding Hamiltonian cycles and longest cycles in outerplanar graphs may be solved in [[linear time]], in contrast to the [[NP-complete]]ness of these problems for arbitrary graphs.

Every maximal outerplanar graph satisfies a stronger condition than Hamiltonicity: it is [[pancyclic graph|node pancyclic]], meaning that for every vertex ''v'' and every ''k'' in the range from three to the number of vertices in the graph, there is a length-''k'' cycle containing ''v''. A cycle of this length may be found by repeatedly removing a triangle that is connected to the rest of the graph by a single edge, such that the removed vertex is not ''v'', until the outer face of the remaining graph has length ''k''.<ref>{{harvtxt|Li|Corneil|Mendelsohn|2000}}, Proposition 2.5.</ref>

A planar graph is outerplanar if and only if each of its biconnected components is outerplanar.<ref name="s79">{{harvtxt|Sysło|1979}}.</ref>