Dirac-type characterizations of graphs without long chordless cycles

We call a chordless path <f>v1v2…vi</f> simplicial if it does not extend into any chordless path <f>v0v1v2…vivi+1</f>. Trivially, for every positive integer k, a graph contains no chordless cycle of length <f>k+3</f> 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 <f>k=1</f> is a classic result of Dirac. [Copyright &y& Elsevier]