Geodesic
Edge cycle extendable graphs
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 2, pp. 373-378

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A graph is edge cycle extendable if every cycle C that is formed from edges and one chord of a larger cycle C⁺ is also formed from edges and one chord of a cycle C' of length one greater than C with V(C') ⊆ V(C⁺). Edge cycle extendable graphs are characterized by every block being either chordal (every nontriangular cycle has a chord) or chordless (no nontriangular cycle has a chord); equivalently, every chord of a cycle of length five or more has a noncrossing chord.
Keywords: cycle extendable graph, chordal graph, chordless graph, minimally 2-connected graph 
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McKee, Terry. Edge cycle extendable graphs. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 2, pp. 373-378. http://geodesic.mathdoc.fr/item/DMGT_2012_32_2_a14/