Geodesic
Strongly pancyclic and dual-pancyclic graphs
Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 1, pp. 5-14

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Say that a cycle C almost contains a cycle C¯ if every edge except one of C¯ is an edge of C. Call a graph G strongly pancyclic if every nontriangular cycle C almost contains another cycle C¯ and every nonspanning cycle C is almost contained in another cycle C⁺. This is equivalent to requiring, in addition, that the sizes of C¯ and C⁺ differ by one from the size of C. Strongly pancyclic graphs are pancyclic and chordal, and their cycles enjoy certain interpolation and extrapolation properties with respect to almost containment. Much of this carries over from graphic to cographic matroids; the resulting 'dual-pancyclic' graphs are shown to be exactly the 3-regular dual-chordal graphs.
Keywords: pancyclic graph, cycle extendable, chordal graph, pancyclic matroid, dual-chordal graph 
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McKee, Terry. Strongly pancyclic and dual-pancyclic graphs. Discussiones Mathematicae. Graph Theory, Tome 29 (2009) no. 1, pp. 5-14. http://geodesic.mathdoc.fr/item/DMGT_2009_29_1_a0/