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.
@article{bwmeta1.element.doi-10_7151_dmgt_1755,
author = {Terry A. McKee},
title = {Pairs Of Edges As Chords And As Cut-Edges},
journal = {Discussiones Mathematicae Graph Theory},
volume = {34},
year = {2014},
pages = {673-681},
zbl = {1303.05102},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_7151_dmgt_1755}
}
Terry A. McKee. Pairs Of Edges As Chords And As Cut-Edges. Discussiones Mathematicae Graph Theory, Tome 34 (2014) pp. 673-681. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_7151_dmgt_1755/
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