Outerplanar Graph - Biconnectivity and Hamiltonicity

Biconnectivity and Hamiltonicity

An outerplanar graph is biconnected if and only if the outer face of the graph forms a simple cycle without repeated vertices. An outerplanar graph is Hamiltonian if and only if it is biconnected; in this case, the outer face forms the unique Hamiltonian cycle. 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-completeness of these problems for arbitrary graphs.

Every maximal outerplanar graph satisfies a stronger condition than Hamiltonicity: it is 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.

A planar graph is outerplanar if and only if each of its biconnected components is outerplanar.

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