It is easy to characterize chordal graphs by every k‐cycle having at least f(k) = k − 3 chords. I prove new, analogous characterizations of the house‐hole‐domino‐free graphs using f(k) = 2⌊(k − 3)/2⌋, and of the graphs whose blocks are trivially perfect using f(k) = 2k − 7. These three functions f(k) are optimum in that each class contains graphs in which every k‐cycle has exactly f(k) chords. The functions 3⌊(k − 3)/3⌋ and 3k − 11 also characterize related graph classes, but without being optimum. I consider several other graph classes and their optimum functions, and what happens when k‐cycles are replaced with k‐paths. © 2010 Wiley Periodicals, Inc. J Graph Theory 68:137‐147, 2011