Chord-set subgraphs of chordal graphs - Mckee, Terry A., (Utilitas Mathematica, 2009)
A chord-set subgraph of a graph consists of all the chords of some cycle of the host graph. Restricting the host to be chordal but not K4 ensures that chord-set subgraphs are connected; restricting the host to be chordal and K4-free ensures that the class of chord-set subgraphs is hereditary. We characterize the chord-set subgraphs of K4-free chordal graphs, generalizing their being precisely the caterpillars among trees and precisely the outerplanar chordal graphs among 2-connected graphs.