Pancyclicity and NP-completeness in planar graphs Original Research Article

A graph is called v-pancyclic if it contains a cycle of length l containing a given vertex v for 3⩽l⩽n, and a graph G is called vertex pancyclic if G is v-pancyclic for all v. In this paper, we show that it is NP-complete to determine whether a 3-connected cubic planar graph is v-pancyclic for given vertex v, it is NP-complete to determine whether a 3-connected cubic planar graph is pancyclic, and it is NP-complete to determine whether a 3-connected planar graph is vertex pancyclic. We also show that every maximal outplanar graph is vertex pancyclic.