Geodesic
The i-chords of cycles and paths
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 607-615

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An i-chord of a cycle or path is an edge whose endpoints are a distance i ≥ 2 apart along the cycle or path. Motivated by many standard graph classes being describable by the existence of chords, we investigate what happens when i-chords are required for specific values of i. Results include the following: A graph is strongly chordal if and only if, for i ∈ 4,6, every cycle C with |V(C)| ≥ i has an (i/2)-chord. A graph is a threshold graph if and only if, for i ∈ 4,5, every path P with |V(P)| ≥ i has an (i -2)-chord.
Keywords: chord, chordal graph, strongly chordal graph, ptolemaic graph, trivially perfect graph, threshold graph 
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McKee, Terry. The i-chords of cycles and paths. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 607-615. http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a0/