Geodesic
Pairs of edges as chords and as cut-edges
Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 4, pp. 673-681.

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Several authors have studied the graphs for which every edge is a chord of a cycle; among 2-connected graphs, one characterization is that the deletion of one vertex never creates a cut-edge. Two new results: among 3-connected graphs with minimum degree at least 4, every two adjacent edges are chords of a common cycle if and only if deleting two vertices never creates two adjacent cut-edges; among 4-connected graphs, every two edges are always chords of a common cycle.
Keywords: cycle, chord, cut-edge 
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McKee, Terry A. Pairs of edges as chords and as cut-edges. Discussiones Mathematicae. Graph Theory, Tome 34 (2014) no. 4, pp. 673-681. http://geodesic.mathdoc.fr/item/DMGT_2014_34_4_a1/

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