The i-chords of cycles and paths
Terry A. McKee · Discussiones Mathematicae Graph Theory · 2012
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.