Dirac-type characterizations of graphs without long chordless cycles

We call a chordless path v1v2…vi simplicial if it does not extend into any chordless path v0v1v2…vivi+1. Trivially, for every positive integer k, a graph contains no chordless cycle of length k+3 or more if each of its nonempty induced subgraphs contains a simplicial path with at most k vertices; we prove the converse. The case of k=1 is a classic result of Dirac.